Relations, Functions, and Their Graphs

Transcription

1 Chapter Relations, Functions, and Their Graphs. Relations and Functions. Linear and Quadratic Functions. Other Common Functions. Variation and Multivariable Functions.5 Transformations of Functions. Combining Functions.7 Inverses of Functions Chapter Project Chapter Summar Chapter Review Chapter Test B the end of this chapter ou should be able to: What if ou visited outer space in a space shuttle? Knowing how much ou weigh on earth, how much would ou weigh miles above earth? B the end of this chapter, ou ll be able to describe and manipulate relations, functions, and their graphs. To calculate our weight in outer space, ou ll need to solve a variation problem like the one on page 58. You ll master this tpe of problem using the definition of Inverse Variation, found on page 5.

2 8 Chapter Introduction This chapter begins with a stud of relations, which are generalizations of the equations in two variables discussed in Chapter, and then moves on to the more specialized topic of functions. As concepts, relations and functions are more abstract, but at the same time far more powerful and useful than the equations studied thus far in this tet. Functions, in particular, lie at the heart of a great deal of the mathematics that ou will encounter from this point on. Leibniz The histor of the function concept serves as a good illustration of how mathematics develops. One of the first people to use the idea in a mathematical contet was the German mathematician and philosopher Gottfried Leibniz ( 7), one of two people (along with Isaac Newton) usuall credited with the development of calculus. Initiall, Leibniz and other mathematicians tended to use the term to indicate that one quantit could be defined in terms of another b some sort of algebraic epression, and this (incomplete) definition of function is often encountered even toda in elementar mathematics. As the problems that mathematicians were tring to solve increased in compleit, however, it became apparent that functional relations between quantities eisted in situations where no algebraic epression defining the function was possible. One eample came from the stud of heat flow in materials, in which a description of the temperature at a given point at a given time was often given in terms of an infinite sum, not an algebraic epression. The result of numerous refinements and revisions of the function concept is the definition that ou will encounter in this chapter, and is essentiall due to the German mathematician Lejeune Dirichlet (85 859). Dirichlet also refined our notion of what is meant b a variable, and gave us our modern understanding of dependent and independent variables, all of which ou will soon encounter. The proof of the power of functions lies in the multitude and diversit of their applications. As ou work through Chapter, pa special attention to how function notation works. A solid understanding of what function notation means is essential to using functions.

3 . Relations and Functions TOPICS. Relations, domain, and range Relations and Functions Section.. Functions and the vertical line test. Function notation and function evaluation. Implied domain of a function 9 TOPIC Relations, Domain, and Range In Chapter we saw man eamples of equations in two variables. An such equation automaticall defines a relation between the two variables present, in the sense that each ordered pair on the graph of the equation relates a value for one variable (namel, the first coordinate of the ordered pair) to a value for the second variable (the second coordinate). Man applications of mathematics involve relating one variable to another, and we will spend much of the rest of this book studing this. DEFINITION Relations, Domain, and Range A relation is a set of ordered pairs. An set of ordered pairs automaticall relates the set of first coordinates to the set of second coordinates, and these sets have special names. The domain of a relation is the set of all the first coordinates, and the range of a relation is the set of all second coordinates. Relations can be described in man different was. We have alread noted that an equation in two variables describes a relation, as the solution set of the equation is a collection of ordered pairs. Relations can also be described with a simple list of ordered pairs (if the list is not too long), with a picture in the Cartesian plane, and b man other means. The following eample demonstrates some of the common was of describing relations and identifies the domain and range of each relation. Relations, Domains, and Ranges EXAMPLE { ( ) ( ) } a. The set R =,,,,,,,, ππ, is a relation consisting of five ordered pairs. { } The domain of R is the set,,,π, as these four numbers appear as first coordinates in the relation. Note that it is not necessar to list the number twice in the domain, even though it appears twice as a first coordinate in the relation.

4 Chapter { } The range of R is the set,,, π, as these are the numbers that appear as second coordinates. Again, it is not necessar to list twice in the range, even though it is used twice as a second coordinate in the relation. The graph of this relation is simpl a picture of the five ordered pairs plotted in the Cartesian plane, as shown below b. The equation + 7 = describes a relation. Using the skills we learned in Chapter, we can graph the solution set below: Unlike the last eample, this relation consists of an infinite number of ordered pairs, so it is not possible to list them all as a set. One of the ordered pairs in the relation is (, ), since ( )+ 7 ()=. The domain and range of this relation are both the set of real numbers, since ever real number appears as both a first coordinate and a second coordinate in the relation.

5 Relations and Functions Section. c. The picture below describes a relation. and Some of the elements of the relation are,,,,.,,,, (,. 758), but this is another eample of a relation with an infinite number of elements so we cannot list all of them. Using interval notation, the domain of this relation is the closed interval, and the range is the closed interval,. [ ] [ ] d. The picture below describes another relation, similar to the last but still different. The shading indicates that all ordered pairs ling inside the rectangle, as well as those actuall on the rectangle, are elements of the relation. [ ] The domain is again the closed interval, and the range is again the closed interval [, ], but this relation is not identical to the last eample. For instance, the ordered pairs (, ) and (., 5. ) are elements of this relation but are not elements of the relation in Eample c. e. Although we will almost never encounter relations in this tet that do not consist of ordered pairs of real numbers, nothing in our definition prevents us from considering more eotic relations. For eample, the set S = {(, ) is the mother of } is a relation among people. Each element of the relation consists of an ordered pair of a mother and her child. The domain of S is the set of all mothers, and the range of S is the set of all people. (Although advances in cloning are occurring rapidl, as of the writing of this tet, no one has et been born without a mother!)

6 Chapter TOPIC Functions and the Vertical Line Test As important as relations are in mathematics, a special tpe of relation, called a function, is of even greater use. DEFINITION Functions A function is a relation in which ever element of the domain is paired with eactl one element of the range. Equivalentl, a function is a relation in which no two distinct ordered pairs have the same first coordinate. Note that there is a difference in the wa domains and ranges are treated in the definition of a function: the definition allows for the two distinct ordered pairs to have the same second coordinate, as long as their first coordinates differ. A picture helps in understanding this distinction: Figure : Definition of Functions The relation on the left in Figure has pairs of points that share the same -value (one such pair is indicated in green). This means that some elements of the range are paired with more than one element of the domain. However, each element of the domain is paired with eactl one element of the range. Thus, this relation is a function. On the other hand, the relation on the right in Figure has pairs of points that share the same -value (one such pair is indicated in red). This means that some elements of the domain are paired with more than one element of the range. This relation is not a function.

7 Relations and Functions Section. EXAMPLE Is the Relation a Function? For each relation in Eample, identif whether the relation is also a function. Solutions: a. The relation in Eample a is not a function because the two ordered pairs, and (, ) have the same first coordinate. If either one of these ordered pairs were deleted from the relation, the relation would be a function b. The relation in Eample b is a function. An two distinct ordered pairs that solve the equation + 7 = have different first coordinates. This can also be seen from the graph of the equation. If two ordered pairs have the same first coordinate, the must be aligned verticall, and no two ordered pairs on the graph of + 7 = have this propert.

8 Chapter c. The relation in Eample c is not a function. To prove that a relation is not a function, it is onl necessar to find two ordered pairs with the same first coordinate, and the pairs,, and ( ) show that this relation fails to be a function. d. The relation in Eample d is also not a function. In fact, we can use the same two ordered pairs as in the previous part to prove this fact. e. Finall, the relation in Eample e also fails to be a function. Think of two people who have the same mother; this gives us two ordered pairs with the same first coordinate: (mother, child ) and (mother, child ). Thus, the relation is not a function. In Eample b, we noted that two ordered pairs in the plane have the same first coordinate onl if the are aligned verticall. We could also have used this criterion to determine that the relation in Eample a is not a function, since the two ordered pairs (, ) and (, ) clearl lie on the same vertical line. This visual method of determining whether a relation is a function, called the vertical line test, is ver useful when an accurate graph of the relation is available. Theorem The Vertical Line Test If a relation can be graphed in the Cartesian plane, the relation is a function if and onl if no vertical line passes through the graph more than once. If even one vertical line intersects the graph of the relation two or more times, the relation fails to be a function. CAUTION! Note that vertical lines that miss the graph of a relation entirel don t prevent the relation from being a function; it is onl the presence of a vertical line that hits the graph two or more times that indicates the relation isn t a function. The net eample illustrates some more applications of the vertical line test.

9 Relations and Functions Section. 5 Functions and the Vertical Line Test EXAMPLE a. The relation = {( ) ( ) ( ) } R,,,,,,,,,, graphed below, is a function. An given vertical line in the plane either intersects the graph once or not at all. b. The relation graphed below is not a function, as there are man vertical lines that intersect the graph more than once. The dashed line is one such vertical line. 8 c. The relation graphed below is a function. In this case, ever vertical line in the plane intersects the graph eactl once. 8

10 Chapter TOPIC Function Notation and Function Evaluation When a function is defined with an equation in two variables, one represents the domain (usuall ) and one represents the range (usuall ). Because functions assign each element of the domain eactl one element of the range, we can solve the equation for. This leads to a special notation for functions, called function notation. DEFINITION Function Notation Suppose a function is represented b an equation in two variables, sa and, and we can solve this equation for, the variable representing the range. We can name the function (frequentl using the letter f ), and write it in function notation b solving the equation for and replacing with f. With the function = as an eample, using function notation we write: f =, which is read f of equals two times, minus three. Function notation can also indicate what to do with a specific value of ; f, read f of tells us to plug the value into the formula given for f. The result is f =, f = 5, which we read f of equals 5. DEFINITION Independent and Dependent Variables Given an equation representing a function = f, we call the independent variable and the dependent variable, since the value of depends on the value of we input into the formula for f. EXAMPLE Function Notation Each of the following equations in and represents a function. Rewrite each one using function notation, and then evaluate each function at =. a. = + b. 7+ = c. 5 = d. = Solutions: a. = + f = + f ( )= + = The equation is alread solved for. To write the function in function notation, replace with f( ). Substitute = and evaluate. This means the point (,) is on the graph of f.

11 Relations and Functions Section. 7 b. 7+ = 7 = = 7 7 = + 7 g= + 7 g( )= ( )+ = 7 c. 5 = h = + 5 h = + 5 ( )= ( ) + = d. = 5 The first step is to solve the equation for the dependent variable. We can name the function anthing at all. Tpical names of functions are f, g, h, etc. We will use g to differentiate this function from the one in part a. 7 Now evaluate g at. The point, is on the graph of g. Again, begin b solving for. To distinguish this function, use a different name. Substitute into the function. The point (, ) is on the graph of h. As usual, the process begins b solving for. = = + j= + j ( )= + ( ) = + = Generall, we avoid using i as a function name, since i also represents the imaginar unit. Substitute and then simplif to evaluate j ( ). This tells us that, is on the graph of j. CAUTION! B far the most common error made when encountering functions for the first time is to think that f() stands for the product of f and. This is entirel wrong! While it is true that parentheses are often used to indicate multiplication, the are also used in defining functions.

12 8 Chapter DEFINITION Argument of a Function In defining a function f, such as f=, the critical idea is the formula. We can use an smbol at all as the variable in defining the formula that we have named f. For instance, f( n)= n, f( z)= z, and f ( $ )= ( $ ) all define eactl the same function. The variable (or smbol) that is used in defining a given function is called its argument, and serves as nothing more than a placeholder. We will not alwas be replacing the arguments of functions with numbers. In man instances, we will have reason to replace the argument of a function with another variable or possibl a more complicated algebraic epression. Keep in mind that this just involves substituting something for the placeholder used in defining the function. EXAMPLE 5 Evaluating Functions Note: The epression in part c. of this eample is called the difference quotient of a function, and is used heavil in calculus. Given the function f =, evaluate: a. f a b. f ( + h) c. Solutions: a. f ( a)= a b. f = + h + h = ( + h+ h ) = + h+ h + + c. f h f h h = h h h + h = h h + h = h = + h f + h f h This is just a matter of replacing with a. Here we replace with + h and simplif the result. We can use the result from above in simplifing this epression. Simplif. Factor out h, so that we can cancel out the h in the denominator. There is one final piece of function notation that is often encountered, especiall in later math classes such as calculus. DEFINITION Domain and Codomain Notation The notation f : A B (read f defined from A to B or f maps A to B ) implies that f is a function from the set A to the set B. The smbols indicate that the domain of f is the set A, and that the range of f is a subset of the set B. In this contet, the set B is often called the codomain of f.

13 Relations and Functions Section. 9 Note that while the notation f : A B implies the domain of f is the entire set A, there is no requirement for the range of f to be all of B. If it so happens that the range of f actuall is the entire set B, f is said to be onto B (or, more formall, to be a surjective function). The net eample illustrates how this notation is tpicall encountered, and also points out some of the subtleties inherent in these notions. EXAMPLE Domain, Codomain, and Range Identif the domain, the codomain, and the range of each of the following functions. [ ) a. f : R R b f = b. g: R, b g= c. h: Z Z b h= d. j: N R b j= Solutions: a. The f : R R portion of the statement tells us that a function f on the real numbers is about to be defined, and that each value of the function will also be a real number. That is, the domain and codomain of f are both R. The f = portion tells us the details of how the function acts. Namel, it returns the square of each real number it is given. Since the square of an real number is nonnegative, and since ever nonnegative real number is the square of some real number, the range of f is the interval [, ). b. The function g is ver similar to the function f in part a. The onl difference is that the notation g: R [, ) tells us in advance that the codomain of g is the nonnegative real numbers. Note that the domain of g is R and the range of g is the same as the range of f. But since the range of g is the same as the codomain of g, the function g is said to be onto, or surjective. This points out that the qualit of being onto depends entirel on how the codomain of the function is specified. If it is no larger than the range of the function, then the function is onto. c. The function h: Z Z b h= has a domain and codomain of Z. But if we think about the result of squaring an given integer (positive or negative), we quickl see that the range of h is the set {,,, 9,, 5,...}. That is, the range of h consists of those integers which are squares of other integers. Since the range of h is not the same as the codomain, h is not onto. d. The function j: N R b j= is one final variation on the squaring function. The action of j is the same as that of the previous three functions, but this time the domain is specified to be the natural numbers (the positive integers) and the codomain is the entire set of real numbers. Since the range of j is the set {,, 9,, 5,...}, which is not the same as the codomain, j is not onto.

14 Chapter TOPIC Implied Domain of a Function Occasionall, the domain of a function is made clear b the function definition. However, it is often up to us to determine the domain, to find what numbers ma be plugged into the function so that the output is real number. In these cases, the domain of the function is implied b the formula defining the function. For instance, an values for the argument of a function that result in division b zero or an even root of a negative number must be ecluded from the domain of that function. EXAMPLE 7 Implied Domain of a Function Determine the domain of the following functions. a. f = 5 b. g= Solutions:. Dom = { }, Ran = { 5,,, 9 }. Dom = {, 5,, 5 }, Ran = {,, }. Dom = π, π,,, { } Ran = {,,, 7 }. Dom = {, }, Ran = {, 8, } 5. Dom = Z, Ran = even integers. Dom = { π}, Ran = Q a. Looking at the formula, we can identif what ma cause the function to be undefined. f = 5 The square root term is defined as long as. Solving this inequalit for, we have. Using interval notation, the domain of the function f is the interval (, ]. b. Again, we first identif potential dangers in the formula for this function. g= The denominator will equal zero whenever =. This tells us that we must eclude = and = from the domain. In interval notation, the domain of g is (, ) (, ) (, ). Eercises For each relation below, describe the domain and range. See Eample.. = {( ) ( ) ( ) ( )} { } R, 5,,,,,, 9. S = (, ), ( 5, ), (, ), ( 5, ) { 7 }. B = {(,, ) (,, ) (, 8, ) (, ) }. A = ( π, ),( π, ),(, ),(, ) { } We can alwas multipl a number b 5, but taking the square root of a negative number is undefined. We can safel substitute an value in the numerator, but we can t let the denominator equal zero. { } 5. T = (, ) Z and =. U = ( π, ) Q

15 Relations and Functions Section. 7. Dom = Z, Ran =,,,, { } 8. Dom = multiples of 5, Ran = multiples of { } { } 7. C =,+ Z 8. D= ( 5, ) Z and Z 9. = 7. + =. =. =. =. = 9. Dom = Ran = R. Dom = Ran = R 5. =. = 7π [ ),. Dom =, Ran = R Dom = R, [ ) Ran =,. Dom = R, Ran = { }. Dom = { }, Ran = R 5. Dom = { }, Ran = R. Dom = R, { } Ran = 7π 7. Dom = [, ], Ran = [, ] Dom = R, Ran =, ( ] 9. Dom = [, ], Ran = [, 5 ]. Dom = [, ], Ran = [, 5 ]. Dom = [, ], Ran = [, ]... Dom = [, ], Ran = [, ]. Dom = All males with siblings, Ran = All people who have brothers. Dom = All parents of girls, Ran = All females { }. V, is the brother of. W, is the daughter of = { } =

16 Chapter 5. Not a function; 5, and, ( ). Function 7. Function 8. Not a function; 5,, and ( ) 9. Not a function; (, ) and (, ). Function. Not a function;, and, ( ). Function. Function. Not a function;, and, 5. Function. Not a function; (, ) and (, ) 7. Function; Dom = (, ) (, ) 8. Not a function 9. Not a function. Function; Dom = R. Function; Dom = (, ) (, ). Not a function. Function; Dom = R. Function; Dom = R 5. Not a function. Function; Dom = [, ) 7. f = + f ( )= 8 Determine which of the relations below is a function. For those that are not, identif two ordered pairs with the same first coordinate. See Eamples and. { ( ) ( )} { ( )} 5. R = 5,,,,,,, 9. S =,,, {(, ),(, ),(, ),(, )} 8. U = ( 5, ), (, ), (, ), (, ) 7. T = { ( )} { } { ( ) ( ) ( )} 9. V =,,,,,, 5,. W =,,,,,,,

17 Relations and Functions Section. 8. f = + f ( )= 9. f = + f ( )= 5. f = 5 f ( )= 5. f = f ( )= f = 9 f ( )= 5. a. b. 9 + c. a + a+ a d. 5. a. b. c. + d. 55. a. 8 b. c. a + a d a. b. + 8 c. a a d a. b. + c. a d. + Determine whether each of the following relations is a function. If it is a function, give the relation s domain. 7. = 8. = 9. + =. =. = +. + =. =. = 5. =. = Rewrite each of the relations below as a function of. Then evaluate the function at =. See Eample = + 8. = 9. + = 5. + = = ( + + 5) 5. = For each function below, determine a. See Eample 5. f, b. f, c. f + a f 5. f = + 5. f = 55. f = + 5. f = f = 5 ( ) 58. f = f =. f = + 5 Determine the difference quotient f + h f See Eample 5c. h. f = 5. t= +., and d. f of each of the following functions.. h =. g= f = 5. f = ( + ) 7. f = 7 8. f = 9. f = 7. f = 58. a. 8+ b. + + c. a + a + + a d. +

18 Chapter 59. a. i b. c. a d. + i 5. a. + 5 b. c. + a + 5 d.. + h 5. + h + h. + h+ ( + ). + h ( + h) Identif the domain, the codomain, and the range of each of the following functions. See Eample. 7. f : R R b f = 7. g: Z Z b g= [ ) = 7. f : Z Z b f = g:, R b g 75. h: N N b h= h: N R b h= Determine the implied domain of each of the following functions. See Eample f = 78. g= h = 8. f = + 8. g= + 8. s= 85. c= 8. h = f = g= f = g= 5. + h h h + h h + h 89. h = h = 7. Dom = Cod = Ran = R 7. Dom = Cod = Z; Ran = {...,,,,...} 7. Dom = Cod = Ran = Z 7. Dom = [, ); Cod = R ; Ran = [, ) 75. Dom = Cod = N; Ran = {, 7, 8,...} 7. Dom = N; Cod = R; Ran =,,,, [, ) 78. R 79. (, ) (, ), 8. [, ) 8. R 8. (, ), 8.,, 8. R 85.,, 8., [ ) 87., 88. R 89. (, ), ( ] 9.,

19 . Linear and Quadratic Functions Section. Linear and Quadratic Functions TOPICS. Linear functions and their graphs. Quadratic functions and their graphs. Maimization/minimization problems T. Maimum/minimum of graphs 5 TOPIC Linear Functions and Their Graphs Much of the net several sections of this chapter will be devoted to gaining familiarit with some of the tpes of functions that commonl arise in mathematics. We will discuss two classes of functions in this section, beginning with linear functions. Recall that a linear equation is an equation whose graph consists of a straight line in the Cartesian plane. Similarl, a linear function is a function whose graph is a straight line. We can define such functions algebraicall as follows. DEFINITION Linear Functions A linear function, sa f, of one variable, sa the variable, is an function that can be written in the form f = m + b, where m and b are real numbers. If m, f = m + b is also called a first-degree function. In the last section, we learned that a function defined b an equation in and can be written in function form b solving the equation for and then replacing with f. This process can be reversed, so the linear function f = m + b appears in equation form as = m+ b, a linear equation written in slope-intercept form. Thus, the graph of a linear function is a straight line with slope m and -intercept, b. As we noted in Section., the graph of a function is a plot of all the ordered pairs that make up the function; that is, the graph of a function f is the plot of all the ordered pairs in the set {(, ) f = }. We have a great deal of eperience in plotting such sets if the ordered pairs are defined b an equation in and, but we have onl plotted a few functions that have been defined with function notation. An function of defined with function notation can be written as an equation in and b replacing f () with, so the graph of a function f consists of a plot of the ordered pairs in the set f {(, ) domain of f }.

20 Chapter Consider the function f = + 5. Figure contains a table of four ordered pairs defined b the function and a graph of the function with the four ordered pairs noted. f Figure : Graph of f= + 5 Again, note that ever point on the graph of the function in Figure is an ordered pair of the form (, f ); we have simpl highlighted four of them with dots. We could have graphed the function f = + 5 b noting that it is a straight line with a slope of and a -intercept of 5. We use this approach in the following eample. EXAMPLE Graphing Linear Functions Graph the following linear functions. a. f = + b. g= Note: A function cannot represent a vertical line (since it fails the vertical line test). Vertical lines can represent the graphs of equations, but not functions. Solutions: a. The function f is a line with a slope of and a -intercept of. To graph the function, plot the ordered pair (, ) and locate another point on the line b moving up units and over to the right unit, giving the ordered pair (, 5). Once these two points have been plotted, connecting them with a straight line completes the process.

21 Linear and Quadratic Functions Section. 7 b. The graph of the function g is a straight line with a slope of and a -intercept of. A linear function with a slope of is also called a constant function, as it turns an input into one fied constant in this case the number. The graph of a constant function is alwas a horizontal line. TOPIC Quadratic Functions and Their Graphs In Section.7, we learned how to solve quadratic equations in one variable. We will now stud quadratic functions of one variable and relate this new material to what we alread know. DEFINITION Quadratic Functions A quadratic function, or second-degree function, of one variable is an function that can be written in the form f = a + b + c, where a, b, and c are real numbers and a. The graph of an quadratic function is a roughl U-shaped curve known as a parabola. We will stud parabolas further in Chapter 9, but in this section we will learn how to graph parabolas as the arise in the contet of quadratic functions. The graph in Figure is the most basic eample of a parabola; it is the graph of the quadratic function f =, and the table that appears alongside the graph contains a few of the ordered pairs on the graph. f 9 8 Figure : Graph of f=

22 8 Chapter DEFINITION Verte and Ais of a Parabola Figure demonstrates two ke characteristics of parabolas: There is one point, known as the verte, where the graph changes direction. Scanning the graph from left to right, it is the point where the graph stops going down and begins to go up (if the parabola opens upward) or stops going up and begins to go down (if the parabola opens downward). Ever parabola is smmetric with respect to its ais, a line passing through the verte dividing the parabola into two halves that are mirror images of each other. This line is also called the ais of smmetr. Ever parabola that represents the graph of a quadratic function has a vertical ais, but we will see parabolas later in the tet that have nonvertical aes. Finall, parabolas can be relativel skinn or relativel broad, meaning that the curve of the parabola at the verte can range from ver sharp to ver flat. We will develop our graphing method b working from the answer backward. We will first see what effects various mathematical operations have on the graphs of parabolas, and then see how this knowledge lets us graph a general quadratic function. To begin, the graph of the function f =, shown in Figure, is the basic parabola. We alread know its characteristics: its verte is at the origin, its ais is the -ais, it opens upward, and the sharpness of the curve at its verte will serve as a convenient reference when discussing other parabolas. Now consider the function g= ( ), obtained b replacing in the formula for f with. We know is equal to when =. What value of results in ( ) equaling? The answer is =. In other words, the point (, ) on the graph of f corresponds to the point (, ) on the graph of g. With this in mind, eamine the table and graph in Figure. g Figure : Graph of g = ( ) Notice that the shape of the graph of g is identical to that of f, but it has been shifted over to the right b units. This is our first eample of how we can manipulate graphs of functions, a topic we will full eplore in Section.5.

23 Linear and Quadratic Functions Section. 9 Now consider the function h obtained b replacing the in with + 7. As with the functions f and g, h =( + 7) is nonnegative for all values of, and onl one value for will return a value of : h( 7)=. Compare the table and graph in Figure with those in Figures and. h Figure : Graph of h = ( + 7) So we have seen how to shift the basic parabola to the left and right: the graph of g= ( h) has the same shape as the graph of f =, but it is shifted h units to the right if h is positive and h units to the left if h is negative. How do we shift a parabola up and down? To move the graph of f = up b a fied number of units, we need to add that number of units to the second coordinate of each ordered pair. Similarl, to move the graph down we subtract the desired number of units from each second coordinate. To see this, consider the table and graphs for the two functions j= + 5 and k= in Figure 5. j k j() k() 8 Figure 5: Graph of j = + 5 and k= Finall, how do we make a parabola skinnier or broader? To make the basic parabola skinnier (to make the curve at the verte sharper), we need to stretch the graph verticall. We can do this b multipling the formula b a constant a greater than to obtain the formula a. Multipling the formula b a constant a that lies between and makes the parabola broader (it makes the curve at the verte flatter). Finall, multipling b a negative constant a turns all of the nonnegative outputs of f into nonpositive outputs, resulting in a parabola that opens downward instead of upward.

24 Chapter Compare the graphs of l= and m = in Figure to the basic parabola f =, which is shown as a green curve. l m l() m() Figure : Graph of I= and m= The following form of a quadratic function brings together all of the above was of altering the basic parabola f =. DEFINITION Verte Form of a Quadratic Function The graph of the function g= a ( h) + k, where a, h, and k are real numbers and a, is a parabola whose verte is at (h, k). The parabola is narrower than f = if a >, and is broader than f = if < a <,. The parabola opens upward if a is positive and downward if a is negative. Where does this definition come from? Consider this construction of the verte form: f = Begin with the basic parabola, with verte (, ). g= ( h) This represents a horizontal shift of h. This step adds a vertical shift of k, making the new g= ( h) + k verte ( hk, ). g= a( h) + k Finall, appl the stretch/compress factor of a. If a is negative, the parabola opens downward. The question now is: given a quadratic function f = a + b + c, how do we deter - mine the location of its verte, whether it opens upward or downward, and whether it is skinnier or broader than the basic parabola? All of this information is available if the equation is in verte form, so we need to convert the formula a + b + c into the form a ( h) + k. It turns out that we can alwas do this b completing the square on the first two terms of the epression.

25 Linear and Quadratic Functions Section. EXAMPLE Graphing Quadratic Functions Note: Finding and plotting the -intercepts is a great wa to see the shape of the function. Sketch the graph of the function f = +. Locate the verte and the -intercepts. Solution: First, identif the verte of the function b completing the square as shown below. First, factor out the leading coefficient of from f = + the first two terms. = ( + )+ Complete the square on the and terms. = = + Because of the in front of the parentheses, this amounts to adding to the function, so we compensate b adding as well. Completing the square places the equation in verte form, and we rewrite the epression +, so the verte is,. + + as The instructions also ask us to identif the -intercepts. An -intercept of the function f is an point on the -ais where f =, so we need to solve the equation + =. This can be done b factoring: + = + = First, divide each term b. ( + ) ( )= Factor into two binomials. =, The Zero-Factor Propert gives us the -intercepts. Therefore, the -intercepts are located at (, ) and (, ). The verte form of the function, f = ( + ) +, tells us that this quadratic opens downward, has its verte at (, ), and is neither skinnier nor broader than the basic parabola. We now also know that it crosses the -ais at and. Putting this all together, we obtain the following graph

26 Chapter In Section.7, we completed the square on the generic quadratic equation to develop the quadratic formula. We can use a similar approach to transform the standard form of a quadratic function into verte form. f = a + b + c b = a a + + c a b a b = + + a b a a + c b b = a + + c a a b ac b = a + + a a As alwas, begin b factoring the leading coefficient a from the first two terms. To complete the square, add the square of half of b inside the parentheses. We need to balance a the equation b subtracting a b a outside the parentheses, then simplif. THEOREM Verte of a Quadratic Function Given a quadratic function f = a + b + c, the graph of f is a parabola with a verte given b: b = a f b b ac b, a, a a. EXAMPLE Using the Verte Formula Note: If the -coordinate of the verte is simple, use substitution to find the -coordinate. If the -coordinate is complicated, use the eplicit formula (the right-hand form in the definition above). Find the verte of the following quadratic functions using the verte formula. a. f = + 8 b. g= + 5 Solutions: a. Begin b using the formula to find the -coordinate of the verte: + 8 = b a () = = At this point, we need to decide how to find the -coordinate. Since the -coordinate b is an integer, substitute it directl into the original equation, finding f a. = + f 8 = Thus, the verte of the graph of f Note that the value of a is. Substitute a and b into the formula and simplif. is ( ),.

27 Linear and Quadratic Functions Section. b. Again, begin b finding the -coordinate of the verte. + 5 = b 5 5 a = = Here, the -coordinate is a fraction, so substituting it into the original equation leads to mess calculations. Instead, use the eplicit formula to find the -coordinate. ac b a = 5 = 7 = 5 Thus, the verte of the graph of g Substitute a and b into the formula and simplif. Substitute a, b, and c into the formula and simplif. is 5 7,. TOPIC Maimization/Minimization Problems Man applications of mathematics involve determining the value (or values) of the variable that return either the maimum or minimum possible value of some function f(). Such problems are called Ma/Min problems for short. Eamples from business include minimizing cost functions and maimizing profit functions. Eamples from phsics include maimizing a function that measures the height of a rocket as a function of time and minimizing a function that measures the energ required b a particle accelerator. If we have a Ma/Min problem involving a quadratic function, we can solve it b finding the verte. Recall that the verte is the onl point where the graph of a parabola changes direction. This means it will be the minimum value of a function (if the parabola opens upward) or the maimum value (if the parabola opens downward). Maimum (Verte) Parabola opening downward Minimum (Verte) Parabola opening upward Figure 7: Maimum/Minimum Values of Quadratic Functions

28 Chapter EXAMPLE Fencing a Garden A farmer plans to use feet of spare fencing material to form a rectangular garden plot against the side of a long barn, using the barn as one side of the plot. How should he split up the fencing among the other three sides in order to maimize the area of the garden plot? Solution: If we let represent the length of one side of the plot, as shown in the diagram below, then the dimensions of the plot are feet b feet. A function representing the area of the plot is A = ( ). If we multipl out the formula for A, we recognize it as a quadratic function A = +. This is a parabola opening downward, so the verte will be the maimum point on the graph of A. Using the verte formula we know that the verte of A is the ordered pair, A ( ) (, or ( 5, A( 5) ). Thus, to maimize area, we should let = 5, ) and so = 5. The resulting maimum possible area, 5 5, or 5 square feet, is also the value A( 5). TOPIC T Maimum/Minimum of Graphs As we ve seen, finding the maimum or minimum possible values of some function f () can be etremel important, and we have a method for doing so when the function is quadratic. But what if we wanted to find the minimum of the function f= + 7 +? One wa is to graph it on a calculator, shown below with the following window settings: Xmin = 5, Xma = 5, Ymin =, Yma =. To find the minimum, press TRACE to access the CALC menu and select : minimum. (If we were tring to find the maimum, we would select :maimum.)

29 Linear and Quadratic Functions Section. 5.. The screen should now displa the graph with the words Left Bound? shown at the bottom. Use the arrows to move the cursor anwhere to the left of where the minimum appears to be and press ENTER. The screen should now sa Right Bound? Use the right arrow to move the cursor to the right of where the minimum appears to be and press ENTER again. The tet should now read Guess? Press ENTER a third time and the - and -values of the minimum will appear at the bottom of the screen... So the minimum is approimatel. 89,. 9. Eercises Graph the following linear functions. See Eample.. f () = 5 +. g=. p = 5. g =. r 7. f = ( ) 8. a = + 8. g=. m = +5. h. q. h = + = 5 9. f () = = + = 5. k =.5 5. w= ( ) ( + ) 8. Graph the following quadratic functions, locating the vertices and -intercepts (if an) accuratel. See Eample.. f = ( ) + 7. g= ( + ) 8. h = F= +. G=. p = + +. q = + +. r=. s= ( ).....

30 Chapter m = + +. n =( + ) ( ) 7. p = f = 9. k=. q = + +. Verte: (, ) No -int. Match the following functions with their graphs.. f = ( 8 ) 7+ a. b. 7. Verte: (, ) No -int. 8. Verte: (, ) -int.: =, = +. f = +. f = f = f = 8 c. d. e. f.. f = Verte: (, ) No -int. 7. f = 8+ g. h. 8. f = ( 5) ( + )+ 5. Verte:, -int.: =, 5. Verte:, -int.: =, 8. Verte: (, ) No -int.. Verte: (, ) No -int.

31 Linear and Quadratic Functions Section. 7. Verte: (, ), -int.: = 5. Verte: (, ) No -int. Solve the following application problems. See Eample. 9. Cind wants to construct three rectangular dog-training arenas sideb-side, as shown, using a total of feet of fencing. What should the overall length and width be in order to maimize the area of the three combined arenas? (Suggestion: let represent the width, as shown, and find an epression for the overall length in terms of.). A rancher has a rectangular piece of sheet metal that is inches wide b feet long. He plans to fold the metal into a three-sided channel and weld two other sheets of metal to the ends to form a watering trough feet long, as shown. How should he fold the metal in order to maimize the volume of the resulting trough?. Verte: (, ) -int.: =, 7. Verte: (, ) No -int. 8. Verte: (, ) -int.: =±. Among all the pairs of numbers with a sum of, find the pair whose product is maimum.. Among all rectangles that have a perimeter of, find the dimensions of the one whose area is largest.. Find the point on the line + = 5 that is closest to the origin. (Hint: Instead of tring to minimize the distance between the origin and points on the line, minimize the square of the distance.). Among all the pairs of numbers (, ) such that + =, find the pair for which the sum of the squares is minimum. feet 5. Find a pair of numbers whose product is maimum if the pair must have a sum of.. Search the Seas cruise ship has a conference room offering unlimited internet access that can hold up to people. Companies can reserve the room for groups of 8 or more. If the group contains 8 people, the compan pas $ per person. The cost per person is reduced b $ for each person in ecess of 8. Find the size of the group that maimizes the income for the owners of the ship and find this income. 9. Verte: (, ) -int.: =,. Verte: (, ) -int.: =. g. e. a. f 5. c. b 7. h 8. d 9. Width of 5 feet, length of feet. Width and length are 5.,. 8,. The dimensions should be 5 inches b inches b feet 5. 8 and 8. 5 and 5. 9 people; $

32 8 Chapter 7.,5 square feet 8. and 9. 5 rooms units 5. 5 cars 5. 5 sets of golf clubs 5. 5 square feet 5. feet 7. The back of George s propert is a creek. George would like to enclose a rectangular area, using the creek as one side and fencing for the other three sides, to create a pasture for his two horses. If he has feet of material, what is the maimum possible area of the pasture? feet of fencing 8. Find a pair of numbers whose product is maimum if two times the first number plus the second number is The total revenue for Thompson's Studio Apartments is given as the function R=., where is the number of rooms rented. What number of rooms rented produces the maimum revenue? 5. The total revenue of Tran s Machiner Rental is given as the function R=., where is the number of units rented. What number of units rented produces the maimum revenue? 5. The total cost of producing a tpe of small car is given b C= , where is the number of cars produced. How man cars should be produced to incur minimum cost? 5. The total cost of manufacturing a set of golf clubs is given b C= 8 +., where is the number of sets of golf clubs produced. How man sets of golf clubs should be manufactured to incur minimum cost? 5. The owner of a parking lot is going to enclose a rectangular area with fencing, using an eisting fence as one of the sides. The owner has feet of new fencing material (which is much less than the length of the eisting fence). What is the maimum possible area that the owner can enclose? For each of the following three problems, use the formula ht ()= t + vt+ h for the height at time t of an object thrown verticall with velocit v (in feet per second) from an initial height of h (in feet). 5. Sitting in a tree, 8 feet above ground level, Sue shoots a pebble straight up with a velocit of feet per second. What is the maimum height attained b the pebble? 8 ft ft/sec

33 Linear and Quadratic Functions Section A ball is thrown upward with a velocit of 8 feet per second from the top of a -foot building. What is the maimum height of the ball? 5. A rock is thrown upward with a velocit of 8 feet per second from the top of a -foot-high cliff. What is the maimum height of the rock? Use a graphing calculator to graph the following quadratic functions. Then determine the verte and -intercepts. 57. f = f = f = 8. f =. f = 5. f = + 8. f = + +. f = feet 5. feet 57. Verte: (, ), -int.: = 8 ± 58. Verte: (, ), -int.: =, 5. f = +. f = +. Verte: (, ),. Verte: (, ),. Verte:,, -int.: =, -int.: = -int.: = ± ı ı ı ı. Verte: (, 5), -int.: = 5, Verte: (, 7), -int.: =, ı ı ı ı. Verte:,, -int.: = ± ı ı ı ı ı ı ı ı 59. Verte: (, ), -int.: =, 8 ı 8 ı ı ı 5. Verte: (5, ), -int.: = 5± ı 8 ı ı ı

34 . Chapter Other Common Functions TOPICS. Functions of the form a n. Functions of the form a n. Functions of the form a n. The absolute value function 5. The greatest integer function. Piecewise-defined functions TOPIC In Section., we investigated the behavior of linear and quadratic functions, but these are just two tpes of commonl occurring functions; there are man other functions that arise naturall in solving various problems. In this section, we will eplore several other classes of functions, building up a portfolio of functions to be familiar with. Functions of the Form a n We alread know what the graph of an function of the form f = a or f = a looks like, as these are, respectivel, simple linear and quadratic functions. What happens to the graphs as we increase the eponent, and consider functions of the form f = a, f = a, etc.? The behavior of a function of the form f= a n, where a is a real number and n is a natural number, falls into one of two categories. Consider the graphs in Figure : f() = f() = f() = 5 Figure : Odd Eponents The three graphs in Figure show the behavior of f = n for the first three odd eponents. Note that in each case, the domain and the range of the function are both the entire set of real numbers; the same is true for higher odd eponents as well. Now, consider the graphs in Figure :

35 Other Common Functions Section. f() = f() = f() = Figure : Even Eponents These three functions are also similar to one another. The first one is the basic parabola we studied in Section.. The other two bear some similarit to parabolas, but are flatter near the origin and rise more steepl for >. For an function of the form f = n where n is an even natural number, the domain is the entire set of real numbers and the range is the interval,. [ ) Multipling a function of the form n b a constant a has the effect that we noticed in Section.. If a >, the graph of the function is stretched verticall; if < a <, the graph is compressed verticall; and if a <, the graph is reflected with respect to the -ais. We can use this knowledge, along with plotting a few specific points, to quickl sketch graphs of an function of the form f= a n. EXAMPLE Functions of the Form a n Sketch the graphs of the following functions. a. f = 5 Solutions: b. g= a. The graph of the function f will have the same basic shape as the function, but compressed verticall because of the factor of. To make the sketch more 5 accurate, calculate the coordinates of a few points on the graph. The graph to the left illustrates that f = 5 f =. 5 and that

36 Chapter b. We know that the function g will have the same shape as the function, but reflected over the -ais because of the factor of. The graph to the left illustrates this. We also plot a few points on the graph of g, namel (, ) and (, ), as a check. TOPIC Functions of the Form a n We could also describe the following functions as having the form a n, where a is a real number and n is a natural number. Once again, the graphs of these functions fall roughl into two categories, as illustrated in Figures and. f ( ) = f ( ) = f ( ) = 5 Figure : Odd Eponents f ( ) = f ( ) = f ( ) = Figure : Even Eponents

37 Other Common Functions Section. As with functions of the form a n, increasing the eponent on functions of the form a sharpens the curve of the graph near the origin. Note that the domain of an n a function of the form f = is (, ) (, ), but that the range depends on n whether n is even or odd. When n is odd, the range is also (, ) (, ), and when n is even the range is (, ). EXAMPLE Functions of the Form a n Sketch the graph of the function f =. Solution: The graph of the function f is similar to that of the function, with two differences. We obtain the formula b multipling b, a negative number between and. So one difference is that the graph of f is the reflection of with respect to the -ais. The other difference is that the graph of f is compressed verticall. With the above in mind, we can calculate the coordinates of a few points (such as, and, ) and sketch the graph of f as shown below., 8,,,, 8,,, TOPIC Functions of the Form a n n Using radical notation, these are functions of the form a, where a is again a real number and n is a natural number. Square root and cube root functions, in particular, are commonl seen in mathematics. Functions of this form again fall into one of two categories, depending on whether n is odd or even. To begin with, note that the domain and range are both the entire set of real numbers when n is odd, and that both are the interval [, ) when n is even. Figures 5 and illustrate the two basic shapes of functions of this form.

38 Chapter f ( ) = f ( ) = f ( ) = 5 Figure 5: Odd Roots f ( ) = f ( ) = f ( ) = Figure : Even Roots At this point, ou ma be thinking that the graphs in Figures 5 and appear familiar. The shapes in Figure 5 are the same as those seen in Figure, but rotated b 9 degrees and reflected with respect to the -ais. Similarl, the shapes in Figure bear some resemblance to those in Figure, ecept that half of the graphs appear to have been erased. This resemblance is no accident, given that n th roots undo n th powers. We will eplore this observation in much more detail in Section.7. TOPIC The Absolute Value Function The basic absolute value function is f =. Note that for an value of, f () is nonnegative, so the graph of f should lie on or above the -ais. One wa to determine its eact shape is to review the definition of absolute value: if = if < This means that for nonnegative values of, f () is a linear function with a slope of, and for negative values of, f () is a linear function with a slope of. Both linear functions have a -intercept of, so the complete graph of f is as shown in Figure 7.

39 Other Common Functions Section. 5 Figure 7: The Absolute Value Function The effect of multipling b a real number a is what we have come to epect: if a >, the graph is stretched verticall; if < a <, the graph is compressed verticall; and if a is negative, the graph is reflected with respect to the -ais. EXAMPLE The Absolute Value Function Sketch the graph of the function f =. Solution: The graph of f will be a verticall stretched version of, reflected over the -ais. As alwas, we can plot a few points to verif that our reasoning is correct. In the graph below, we have plotted the values of f and f.

40 Chapter TOPIC 5 The Greatest Integer Function DEFINITION The Greatest Integer Function =, is a function commonl encountered in The greatest integer function, f computer science applications. It is defined as follows: the greatest integer of is the largest integer less than or equal to. For instance,. = and 9. = (note that is the largest integer to the left of.9 on the real number line). Careful stud of the greatest integer function reveals that its graph must consist of intervals where the function is constant, and that these portions of the graph must be separated b discrete jumps, or breaks, in the graph. For instance, an value for chosen from the interval [, ) results in f =, but f =. Similarl, an value for chosen from the interval [, ) results in f =, but f ( ) =. Our graph of the greatest integer function must somehow indicate this repeated pattern of jumps. In cases like this, it is conventional to use an open circle on the graph to indicate that the function is either undefined at that point or is defined to be another value. Closed circles are used to emphasize that a certain point reall does lie on the graph of the function. With these conventions in mind, the graph of the greatest integer function appears in Figure 8. Figure 8: The Greatest Integer Function TOPIC Piecewise-Defined Functions There is no rule stating that a function needs to be defined b a single formula. In fact, we have worked with such a function alread; in evaluating the absolute value of, we use one formula if is greater than or equal to and a different formula if is less than. Obviousl, we can t have two rules govern the same input, but we can have multiple formulas on separate pieces of a function s domain.

41 Other Common Functions Section. 7 DEFINITION Piecewise-Defined Function A piecewise-defined function is a function defined in terms of two or more formulas, each valid for its own unique portion of the real number line. In evaluating a piecewisedefined function f at a certain value for, it is important to correctl identif which formula is valid for that particular value. Piecewise-Defined Function Note: Alwas pla close attention to the boundar points of each interval. Remember that onl one rule applies at each point. EXAMPLE Sketch the graph of the function f ( )= if. if > Solution: The function f is a piecewise function with a different formula for two intervals. To graph f, graph each portion separatel, making sure that each formula is applied onl on the appropriate interval. ( ] The function f is a linear function on the interval, and a quadratic function on the interval,. The complete graph appears below, with the points f in particular. ( )= and f = noted (,) (,) (,) (,) Note the use of a closed circle at, to emphasize that this point is part of the graph, and the use of an open circle at (, ) to indicate that this point is not part of the graph. That is, the value of f is, not.

42 8 Chapter. Eercises. Sketch the graphs of the following functions. Pa particular attention to intercepts, if an, and locate these accuratel. See Eamples through.. f =. g=. F=.. h = 5. p =. q = 7. G= 8. k= 9. G =.. H= 5.. r=. p = 5.. W=. k= 9 5. h =. S= 7. d= 5 8. f =. 9. r=. s=. t= 7.. f =. P=. m =

43 Other Common Functions Section. 9. if < 5. f = if. g ( )= if if > 7. if < 7. r= if > + if < 8. p = if < if q = if Z if Z. s( )= if < if 9.. v = if if < or >. h = if < if if Z. M= if Z if. u = if >

44 5 Chapter 5. f. d 7. g 8. a 9. h. e. b. c Match the following functions to their graphs. 5. f =. f = f = 7 a. b. c. d. 8. f = f= e. f.. f = 7 if. f = 5 if > g. h. if <. f = if

45 . Variation and Multivariable Functions Section. Variation and Multivariable Functions TOPICS. Direct variation. Inverse variation. Joint variation. Multivariable functions 5 TOPIC Direct Variation A number of natural phenomena ehibit the mathematical propert of variation: one quantit varies (or changes) as a result of a change in another quantit. One eample is the electrostatic force of attraction between two oppositel charged particles, which varies in response to the distance between the particles. Another eample is the distance traveled b a falling object, which varies as time increases. Of course, the principle underling variation is that of functional dependence; in the first eample, the force of attraction is a function of distance, and in the second eample the distance traveled is a function of time. We have now gained enough familiarit with functions that we can define the most common forms of variation. DEFINITION Direct Variation We sa that varies directl as the n th power of (or that is proportional to the n th power of ) if there is a nonzero constant k (called the constant of proportionalit) such that = k n. Man variation problems involve determining what, eactl, the constant of proportionalit is in a given situation. This can be easil done if enough information is given about how the various quantities in the problem var with respect to one another, and once k is determined man other questions can be answered. The following eample illustrates the solution of a tpical direct variation problem.

46 5 Chapter EXAMPLE Direct Variation Hooke s Law sas that the force eerted b the spring in a spring scale varies directl with the distance that the spring is stretched. If a 5-pound mass suspended on a spring scale stretches the spring inches, how far will a -pound mass stretch it? Solution: The first equation tells us that F = k, where F represents the force eerted b the spring and represents the distance that the spring is stretched. When a mass is suspended on a spring scale (and is stationar), the force eerted upward b the spring must equal the force downward due to gravit, so the spring eerts a force of 5 pounds when a 5-pound mass is suspended from it. So the second sentence tells us that Weight in Pounds inches 5= k, or k = 5. We can now answer the question: 5 = =. 5 5 pounds So the spring stretches 5. inches when a -pound mass is suspended from it. TOPIC Inverse Variation In man situations, an increase in one quantit results in a corresponding decrease in another quantit, and vice versa. Again, this is a natural illustration of a functional relationship between quantities, and an appropriate name for this tpe of relationship is inverse variation. DEFINITION Inverse Variation We sa that varies inversel as the n th power of (or that is inversel proportional to the n th power of ) if there is a nonzero constant k such that k =. n

47 Variation and Multivariable Functions Section. 5 The method of solving an inverse variation problem is identical to that seen in the first eample. First, write an equation that epresses the nature of the relationship (including the as-et-unknown constant of proportionalit). Second, use the given information to determine the constant of proportionalit. Third, use the knowledge gained to answer the question. EXAMPLE Inverse Variation The weight of a person, relative to the Earth, is inversel proportional to the square of the person s distance from the center of the Earth. Using a radius for the Earth of 7 kilometers, how much does a 8-pound man weigh when fling in a jet 9 kilometers above the Earth s surface? Solution: If we let W stand for the weight of a person and d the distance between the person and the Earth s center, the first sentence tells us that W = k d. 7 km 9 km The second sentence gives us enough information to determine k. Namel, we know that W = 8 (pounds) when d = 7 (kilometers). Solving the equation for k and substituting in the values that we know, we obtain 9 k = Wd =( 8)( 7) 7.. When the man is 9 kilometers above the Earth s surface, we know d = 79, so the man s weight while fling is W = ( 8 )( 7 ) ( 79) = pounds. Fling is not, therefore, a terribl effective wa to lose weight.

48 5 Chapter TOPIC Joint Variation In more complicated situations, it ma be necessar to identif more than two variables and to epress how the variables relate to one another. And it ma ver well be the case that one quantit varies directl with respect to some variables and inversel with respect to others. For instance, the force of gravitational attraction F between two bodies of mass m and mass m varies directl as the product of the masses and kmm inversel as the square of the distance between the masses: F =. d When one quantit varies directl as two or more other quantities, the word jointl is often used. DEFINITION Joint Variation We sa that z varies jointl as and (or that z is jointl proportional to and ) if there is a nonzero constant k such that z= k. If z varies jointl as the n th power of and the m th power of, we write z= k n m. EXAMPLE Joint Variation The volume of a right circular clinder varies jointl as the height and the square of the radius. Epress this relationship in equation form. Solution: This simple problem merel asks for the form of the variation equation. If we let V stand for the volume of a right circular clinder, r for its radius, and h for its height, we would write V = krh. r h Of course, we are alread familiar with this volume formula and know that the constant of proportionalit is actuall π, so we could provide more information and write V =πr h.

49 Variation and Multivariable Functions Section. 55 TOPIC Multivariable Functions The topic of variation provides an ecellent opportunit to introduce functions that depend on two or more arguments. Eamples abound in both pure and applied mathematics, and as ou progress through calculus and later math classes ou will encounter such functions frequentl. Consider again how the force of gravit F between two objects depends on the masses m and m of the objects and the distance d between them: F km m d =. If we change an of the three quantities m, m, and d, the force F changes in response. A slight etension of our familiar function notation leads us then to epress F as a function of m, m, and d and to write kmm F( m, m, d)=. d In fact, we can be a bit more precise and replace the constant of proportionalit k with G, the Universal Gravitational Constant. Through man measurements in man different eperiments, G has been determined to be approimatel 7. Nm/kg (N stands for the unit of force called the Newton; Newton of force gives a mass of kg an acceleration of m/s ). If we use this value for G, we must be sure to measure the masses of the objects in kilograms and the distance between them in meters. The net eample illustrates an application of this function to which all of us on Earth can relate. EXAMPLE Finding the Force of Gravit Determine the approimate force of gravitational attraction between the Earth and the Moon. Solution: The mass of the Earth is approimatel. kg and the mass of the Moon is approimatel 7. kg. The distance between these two bodies varies, but on average it is 8. 8 m. Using function notation, we would write 8 F (., 7., 8. )= ( 7. )(. )( 7. ) 8 ( 8. ) =. N. It is this force of mutual attraction that keeps the Moon in orbit about the Earth.

50 5 Chapter Eercises Mathematical modeling is the process of finding a function that describes how quantities or variables relate to one another. The function is called the mathematical model. Find the mathematical model for each of the following verbal statements.. A = kbh. V = kr. W = k d. P = k V. A varies directl as the product of b and h.. V varies directl as the product of four-thirds and r cubed.. W varies inversel as d squared.. P varies inversel as V. 5. r varies inversel as t.. S varies directl as the product of four and r squared. 7. varies jointl as the cube of and the square of z. 8. a varies jointl as the square of b and inversel as c. k 5. r = t. S = kr 7. = k z kb 8. a = c 9. = 8 5. =.. =. = 975. =.75. =. 5. z = 8. z = 7. z = 8. z =, feet. 89 CDs Solve the following variation problems. See Eamples,, and. 9. Suppose that varies directl as the square root of, and that = when =. What is when =?. Suppose that varies inversel as the cube of, and that =. 5 when =. What is when = 5?. Suppose that varies directl as the cube root of, and that = 75 when = 5. What is when = 8?. Suppose that is proportional to the 5 th power of, and that = 9 when =. What is when = 5?. Suppose that varies inversel as the square of, and that = when =. What is when = 8?. Suppose that is inversel proportional to the th power of, and that =.5 when =. What is when =? 5. z varies directl as the square of and inversel as. If z = when = and = 7, what value does z have when = and =?. Suppose that z varies jointl as the square of and the cube of, and that z = 78 when = and =. What is z when = and =? 7. Suppose that z is jointl proportional to and, and that z = 9 when =.5 and =. What is z when =.8 and = 7? 8. Suppose that z is jointl proportional to and the cube of, and that z = 988 when = and =. What is z when = 7 and = 8? 9. The distance that an object falls from rest, when air resistance is negligible, varies directl as the square of the time. A stone dropped from rest travels feet in the first seconds. How far does it travel in the first seconds?. A record store manager observes that the number of CDs sold seems to var inversel as the price per CD. If the store sells 8 CDs per week when the price per CD is $5.99, how man does he epect to sell if he lowers the price to $.99?

51 Variation and Multivariable Functions Section pounds..7 meters. amps 5..5 centimeters. A person s Bod Mass Inde (BMI) is used b phsicians to determine if a patient s weight falls within reasonable guidelines relative to the patient s height. The BMI varies directl as a person s weight in pounds and inversel as the square of a person s height in inches. Given that a -foot-tall man weighing 8 pounds has a BMI of., what is the BMI of a woman weighing pounds with a height of 5 feet inches?. The force necessar to keep a car from skidding as it travels along a circular arc varies directl as the product of the weight of the car and the square of the car s speed, and inversel as the radius of the arc. If it takes pounds of force to keep a -pound car moving 5 miles per hour on an arc whose radius is 75 feet, how man pounds of force would be required if the car were to travel miles per hour?. If a beam of width w, height h, and length l is supported at both ends, the maimum load that the beam can hold varies directl as the product of the width and the square of the height, and inversel as the length. A given beam meters long with a width of centimeters and a height of 5 centimeters can hold a load of kilograms when the beam is supported at both ends. If the supports are moved inward so that the effective length of the beam is shorter, the beam can support more load. What should the distance between the supports be if the beam has to hold a load of kilograms? w = cm l = m h = 5 cm lb r = 75 ft. In a simple electric circuit connecting a batter and a light bulb, the current I varies directl as the voltage V but inversel as the resistance R. When a.5-volt batter is connected to a light bulb with resistance. ohms ( Ω ), the current that travels through the circuit is 5 amps. Find the current if the same light bulb is connected to a -volt batter. 5. The amount of time it takes for water to flow down a drainage pipe is inversel proportional to the square of the radius of the pipe. If a pipe of radius cm can empt a sink in 5 seconds, find the radius of a pipe that would allow the sink to drain completel in seconds.

52 58 Chapter. 5 inches 7..5 inches 8. 7 hot dogs in...5 pounds. 9 watts w. BMI ( wh, )= 7 h s. P ( s, e)= e. V(r, h) =.r h. The perimeter of a square varies directl as the length of the side of a square. If the perimeter of a square is 8 inches when one side is 77 inches, what is the perimeter of a square when the side is inches? 7. The circumference of a circle varies directl as the diameter. A circular pizza slice has a length of.5 inches when the circumference of the pizza is.8 inches. What would the circumference of a pizza be if the pizza slice has a length of 5.5 inches? 8. A hot dog vendor has determined that the number of hot dogs she sells a da is inversel proportional to the price she charges. The vendor wants to decide if increasing her price b 5 cents will drive awa too man customers. On average, she sells 8 hot dogs a da at a price of $.5. How man hot dogs can she epect to sell if the price is increased b 5 cents? 9. The surface area of a right circular clinder varies directl as the sum of the radius times the height and the square of the radius. With a height of 8 in. and a radius of 7 in., the surface area of a right circular clinder is 99 in. What would the surface area be if the height equaled 5 in. and the radius equaled. in.?. The gravitational force, F, between an object and the Earth is inversel proportional to the square of the distance from the object to the center of the Earth. If an astronaut weighs 9 pounds on the surface of the Earth, what will this astronaut weigh miles above the Earth? Assume that the radius of the Earth is miles.. In an electrical schematic, the voltage across a load is directl proportional to the power used b the load but inversel proportional to the current through the load. If a computer is connected to a wall outlet and the computer needs 8 volts to run and absorbs 5 watts of power, the current through the computer is amps. Find the power absorbed b the computer if the same 8-volt computer is attached to a circuit with a loop current of.5 amps. Epress the indicated quantities as functions of the other variables. See Eample.. A person s Bod Mass Inde (BMI) varies directl as a person s weight in pounds and inversel as the square of a person s height in inches. Given that a -foot-tall man weighing 8 pounds has a BMI of., epress BMI as a function of weight (w) and height (h).. The electric pressure varies directl as the square of the surface charge densit (s) and inversel as the permittivit (e). If the surface charge densit is coulombs per unit area and the free space permittivit equals, the pressure is equal to N/m. Epress the electric pressure as a function of surface charge densit and permittivit.. The volume of a right circular clinder varies directl as the radius squared times the height of the clinder. If the radius is 7 and the height is, the volume is equal to 5.. Determine the epression of the volume of a right circular clinder.

53 Variation and Multivariable Functions Section b. a = ; 7 b 7. a = 9 ; b 8. a = ; c 9. a = bc; 8 b. a = 7 ; c 8. P =.5g; $. I. F i = 9 ;.5 fc d d = 5 ; cm 9. V =.r h; 9. in. 5. V. F = 8 ; cm P ab = c ; l 7. R =. 9 ; d 7.8 ohms Find an equation for the relationship given and then use the equation to find the unknown value. 5. The variable a is proportional to b. If a = 5 when b = 9, what is a when b =?. The variable a varies directl as b. If a = when b = 9, what is a when b = 7? 7. The variable a varies directl as the square of b. If a = 9 when b =, what is a when b =? 8. The variable a is proportional to the square of b and varies inversel as the square root of c. If a = 8 when b = and c =, what is a when b = and c = 9? 9. The variable a varies jointl as b and c. If a = when b = and c = 5, what is the value of a when b = and c =?. The variable a varies directl as the cube of b and inversel as c. If a = 9 when b = and c = 7, what is the value of a when b = and c =?. The price of gasoline purchased varies directl with the number of gallons of gas purchased. If gallons of gas are purchased for $., what is the price of purchasing gallons?. The illumination, I, of a light source varies directl as the intensit, i, and inversel as the square of the distance, d. If a light source with an intensit of 5 cp (candlepower) has an illumination of fc (foot-candles) at a distance of 5 feet, what is the illumination at a distance of feet? 5 cp 5 ft ft. The force eerted b a spring varies directl with the distance that the spring is stretched. A hanging spring will stretch 9 cm if a weight of 5 grams is placed on the end of the spring. How far will the spring stretch if the weight is increased to grams?. The volume of a clinder varies jointl as its height and the square of its radius. If a clinder has the measurements V =. cubic inches, r = inches, and h = inches, what is the volume of a clinder that has a radius of inches and a height of 8 inches? 5. The volume of a gas in a storage container varies inversel as the pressure on the gas. If the volume is cubic centimeters under a pressure of 8 grams, what would be the volume of the gas if the pressure was decreased to grams? Pressure 8 g cm. F is jointl proportional to a and b and varies inversel as c. If F = when a =, b = 5, and c =, what is the value of F when a =, b =, and c =? 7. The resistance of a wire varies directl as its length and inversel as the square of the diameter. When a wire is 5 feet long and has a diameter of.5 in., it has a resistance of ohms. What is the resistance of a wire that is feet long and has a diameter of.5 in.?

54 Chapter Transformations of Functions TOPICS.5.. Shifting, reflecting, and stretching graphs Smmetr of functions and equations TOPIC Shifting, Reflecting, and Stretching Graphs Much of the material in this section was introduced in Section., in our discussion of quadratic functions. You ma want to review the was in which the basic quadratic function f = can be shifted, stretched, and reflected as ou work through the more general ideas here. THEOREM Horizontal Shifting/ Translation Let f () be a function, and let h be a fied real number. If we replace with h, we obtain a new function g() = f ( h). The graph of g has the same shape as the graph of f, but shifted to the right b h units if h > and shifted to the left b h units if h <. Horizontal Shifting/ Translation EXAMPLE Sketch the graphs of the following functions. a. f = ( + ) b. g= Note: Solutions: Begin b identifing the underling a. function that is being shifted. The basic function being shifted is. Begin b drawing the basic cubic shape f (the shape of = ). Since is replaced b +, the graph of f b units. Note, for eample, that, on the graph. is the graph of shifted to the left is one point

55 Transformations of Functions Section.5 b. g The basic function being shifted is. Start b graphing the basic absolute value function. The graph of g= has the same shape, but shifted to the right b units. Note, for eample, that (, ) lies on the graph of g. CAUTION! The minus sign in the epression h is critical. When ou see an epression in the form + h ou must think of it as ( h). Consider a specific eample: replacing with 5 shifts the graph 5 units to the right, since 5 is positive. Replacing with + 5 shifts the graph 5 units to the left, since we have actuall replaced with 5. THEOREM Vertical Shifting/ Translation Let f () be a function whose graph is known, and let k be a fied real number. The graph of the function g= f + k is the same shape as the graph of f, but shifted upward if k > and downward if k <. EXAMPLE Vertical Shifting/ Translation Sketch the graphs of the following functions. a. f = + b. g=

56 Chapter Note: As before, begin b identifing the basic function being shifted. Solutions: a. f The basic function being shifted is. The graph of f= + is the graph of = shifted up units. Note that this doesn t change the domain. However, the range is affected; the range of f is,,. b. The basic function being shifted is. Begin b graphing the basic cube root shape. g To graph g=, we shift the graph of = down b units. EXAMPLE Horizontal and Vertical Shifting Sketch the graph of the function f = + +. Note: In this case, it doesn t matter which shift we appl first. However, when functions get more complicated, it is usuall best to appl horizontal shifts before vertical shifts. Solution: f The basic function being shifted is. Begin b graphing the basic square root shape. In f we have replaced with +, so shift the basic function units left. Then shift the resulting function unit up.

57 Transformations of Functions Section.5 THEOREM Reflecting with Respect to the Aes Given a function f :. The graph of the function g respect to the -ais. = f is the reflection of the graph of f with. The graph of the function g= f respect to the -ais. is the reflection of the graph of f with In other words, a function is reflected with respect to the -ais b multipling the entire function b, and reflected with respect to the -ais b replacing with. EXAMPLE Reflecting with Respect to the Aes Sketch the graphs of the following functions. a. f = b. g= Note: We state that a function is reflected with respect to particular ais. Visuall, this means the function is reflected over (across) that ais. Solutions: a. f To graph f=, begin with the graph of the basic parabola =. The entire function is multiplied b, so reflect the graph over the -ais, resulting in the original shape turned upside down. Note that the domain is still the entire real line, but the range of f is the interval,. ( ] b. g To graph g=, begin b graphing =, the basic square root. In g, has been replaced b, so reflect the graph with respect to the -ais. Note that this changes the domain but not the range. The domain of g is the interval, and the range is,. ( ] [ )

58 Chapter THEOREM Vertical Stretching and Compressing Let f () be a function and let a be a positive real number.. The graph of the function g= af is stretched verticall compared to the graph of f if a >.. The graph of the function g af the graph of f if < a <. = is compressed verticall compared to EXAMPLE 5 Vertical Stretching and Compressing Sketch the graphs of the following functions. a. f = b. g = 5 Note: When graphing stretched or compressed functions, it ma help to plot a few points of the new function. Solutions: a. f Begin with the graph of. The shape of f is similar to the shape of but all of the -coordinates have been multiplied b the factor of, and are consequentl much smaller. 8 b. 8 g Begin with the graph of the absolute value function. In contrast to the last eample, the graph of g= 5 is stretched compared to the standard absolute value function. Ever second coordinate is multiplied b a factor of 5.

59 Transformations of Functions Section.5 5 If the function g is obtained from the function f b multipling f b a negative real number, think of the number as the product of and a positive real number (namel, its absolute value). This is a simple eample of a function going under multiple transformations. When dealing with more complicated functions, undergoing numerous transformations, we need a procedure for untangling the individual transformations in order to find the correct graph. PROCEDURE Order of Transformations If a function g has been obtained from a simpler function f through a number of transformations, g can be understood b looking for transformations in this order:. horizontal shifts. stretching and compressing. reflections. vertical shifts Consider, for eample, the function g= + +, which has been built up from the basic square root function through a variet of transformations.. First, has been transformed into + b replacing with +, and we know that this corresponds graphicall to a shift to the left of unit.. Net, the function + has been multiplied b to get the function +, and we know that this has the effect of stretching the graph of + verticall.. The function + has then been multiplied b, giving us +, and the graph of this is the reflection of + with respect to the -ais.. Finall, the constant has been added to +, shifting the entire graph upward b units. These transformations are illustrated, in order, in Figure, culminating in the graph of g= Figure : Building the Graph of g= + +

60 Chapter EXAMPLE Order of Transformations Sketch the graph of the function f =. Solution: The basic function that f is similar to is. Following the order of transformations:. If we replace b + (shifting the graph units to the left), we obtain the function, which is closer to what we want. +. There does not appear to be an stretching or compressing transformation.. If we replace b, we have + =, which is equal to f. This reflects the graph of with respect to the -ais. +. Since we have alread found f, we know there is no vertical shift. The entire sequence of transformations is shown below, ending with the graph of f. + Note: An alternate approach to graphing f = is to rewrite the function in the form f =. In this form, the graph of f is the graph of shifted two units to the right, and then reflected with respect to the -ais. The result is the same, as ou should verif. Rewriting an equation in a different form never changes its graph.

61 Transformations of Functions Section.5 7 TOPIC Smmetr of Functions and Equations We know that replacing with reflects the graph of a function with respect to the -ais, but what if f ( )= f? In this case the original graph is the same as the reflection! This means the function f is smmetric with respect to the -ais. DEFINITION -ais Smmetr The graph of a function f has -ais smmetr, or is smmetric with respect to the -ais, if f f = for all in the domain of f. Such functions are called even functions. Figure : A Function with -Ais Smmetr Functions whose graphs have -ais smmetr are called even functions because polnomial functions with onl even eponents form one large class of functions 8 with this propert. Consider the function f = This function is a polnomial of four terms, all of which have even degree. If we replace with and simplif the result, we obtain the function f again: 8 f ( )= 7( ) 5( ) + ( ) 8 = f = Be aware, however, that such polnomial functions are not the onl even functions. We will see man more eamples as we proceed. There is another class of functions for which replacing with results in the eact negative of the original function. That is, f ( )= f for all in the domain, and this means changing the sign of the -coordinate of a point on the graph also changes the sign of the -coordinate. (, ) What does this mean geometricall? Suppose f is such a function, and that f is a point on the graph of f. If we change the sign of both coordinates, we obtain a new point that is the original point reflected through the origin (we can also think of this as reflected over the -ais, then the -ais).

62 8 Chapter For instance, if (, f ) lies in the first quadrant, (, f ) lies in the third, and if (, f ) lies in the second quadrant, (, f ) lies in the fourth. But since f ( )= f, the point (, f ) can be rewritten as (, f( ) ). Written in this form, we know that (, f( ) ). is a point on the graph of f, since an point of the form (?, f (?)) lies on the graph of f. So a function with the propert = has a graph that is smmetric with respect to the origin. f f DEFINITION Origin Smmetr The graph of a function f has origin smmetr, or is smmetric with respect to the origin, if f f = for all in the domain of f. Such functions are called odd functions. Figure : A Function with Origin Smmetr As ou might guess, such functions are called odd because polnomial functions with onl odd eponents serve as simple eamples. For instance, the function f = + 8 is odd: f ( )= ( ) + 8( ) = ( )+ 8( ) = 8 = f As far as functions are concerned, -ais and origin smmetr are the two principal tpes of smmetr. What about -ais smmetr? It is certainl possible to draw a graph that displas -ais smmetr; but unless the graph lies entirel on the -ais, such a graph cannot represent a function. Wh not? Draw a few graphs that are smmetric with respect to the -ais, then appl the Vertical Line Test to these graphs. In order to have -ais smmetr, if (, ) is a point on the graph, then (, ) must also be on the graph, and thus the graph can not represent a function. This brings us back to relations. Recall that an equation in and defines a relation between the two variables. There are three principal tpes of smmetr that equations can possess.

63 Transformations of Functions Section.5 9 DEFINITION Smmetr of Equations We sa that an equation in and is smmetric with respect to:. The -ais if replacing with results in an equivalent equation. The -ais if replacing with results in an equivalent equation. The origin if replacing with and with results in an equivalent equation Knowing the smmetr of a function or an equation can serve as a useful aid in graphing. For instance, when graphing an even function it is onl necessar to graph the part to the right of the -ais, as the left half of the graph is the reflection of the right half with respect to the -ais. Similarl, if a function is odd, the left half of its graph is the reflection of the right half through the origin. EXAMPLE 7 Smmetr of Equations Note: If ou don t know where to begin when sketching a graph, plotting points often helps ou understand the basic shape. Sketch the graphs of the following relations, making use of smmetr. a. f = Solutions: a. b. g= c. = This relation is a function, one that we alread graphed in Section.. Note that it is indeed an even function and ehibits -ais smmetr: f( ) = ( ) = = f b. While we do not et have the tools to graph general polnomial functions, we can obtain a good sketch of g=. (verif this). First, g is odd: g = g If we calculate a few values, such as g=, g = 8, g ()=, and g=, and then reflect these through the origin, we get a good idea of the shape of g.

64 7 Chapter c. 8 The equation = is not a function, but it is a relation in and that has -ais smmetr. If we replace with and simplif the result, we obtain the original equation: = ( ) = The upper half of the graph is the function =, so drawing this and its reflection gives us the complete graph of =. Summar of Smmetr The first column in the table below summarizes the behavior of a graph in the Cartesian plane if it possesses an of the three tpes of smmetr we covered. If the graph is of an equation in and, the algebraic method in the second column can be used to identif the smmetr. The third column gives the algebraic method used to identif the tpe of smmetr if the graph is that of a function f. Finall, the fourth column contains an eample of each tpe of smmetr. A graph is smmetric with respect to: If the graph is of an equation in and, the equation is smmetric with respect to: If the graph is of a function f (), the function is smmetric with respect to: Eample: The -ais if whenever the point (, ) is on the graph, the point (, ) is also on the graph. The -ais if replacing with results in an equivalent equation. The -ais if f f =. We sa the function is even. (, ) (, ) The -ais if whenever the point (, ) is on the graph, the point (, ) is also on the graph. The -ais if replacing with results in an equivalent equation. Not applicable (unless the graph consists onl of points on the -ais). (, ) (, ) The origin if whenever the point (, ) is on the graph, the point (, ) is also on the graph. The origin if replacing with and with results in an equivalent equation. The origin if f f =. We sa the function is odd. (, ) (, )

65 Transformations of Functions Section.5 7. f =. f =. f =. f = 5. f =. f = 7. f = 8. f = 9. f =. f =. f =. f =.. 5. Dom = Ran = R Dom = R, Ran =, [ ) Eercises For each function or graph below, determine the basic function that has been shifted, reflected, stretched, or compressed.. f = ( ) +. f = + 5. f = +. f = + 5. f= + 5. f = ( + ) + 7. f = + 8. f = ( + ) Dom = R, Ran =, ( ] Dom = [ ) Ran = [, ),, 7. Dom = R, Ran =, [ ) 8. Dom = Ran = R 9. Dom =,, Ran =, ( ] [ ). Dom = R, Ran =, [ )

66 7 Chapter.... Dom = (, ) (, ), Ran =, Dom = (, ) (, ), Ran = (, ), Dom = (, ) (, ), Ran = (, ) (, ) Sketch the graphs of the following functions b first identifing the more basic functions that have been shifted, reflected, stretched, or compressed. Then determine the domain and range of each function. See Eamples through.. f = ( + ). G= 5. p = ( + ) +. g= + 7. q = 9. s=. F=. v =. f = 5. b = g= + 8. r= + +. w= ( ). k= +. R= 7. S= 9. h =. W=. W=. S = +. V= +. g= + 9 (Hint: Find a better wa to write the function.) Write a formula for each of the functions described below. 5. Use the function g() =. Move the function units to the left and units down. 7. Use the function g() =. Reflect the function across the -ais and move it units up. 5.. ( ] [ ) Dom =,, Ran =, Dom = Ran = R Dom = R, ( ] Ran =, Use the function g() =. Move the function units to the right and units up. Dom = Ran = R Dom =,,, Ran = (, ), 9.. Dom = R, Ran =, [ ) Dom = R, Ran =, ( ].. 8. Use the function g() =. Move the function units to the right and reflect across the -ais. Dom = R, Ran =, ( ] Dom = (, ) (, ), Ran =,.. Dom =, Ran =, [ ), ( ] Dom = R, Ran =, [ )

67 Transformations of Functions Section f = ( + ). f = ( ) + 7. f = + 8. f = ( ) = ( + ) 9. f = ( + ). f = ( ) + 9. Use the function g=. Move the function unit to the left and reflect across the -ais.. Use the function g=. Move the function units to the right and units up.. Use the function g=. Move the function 5 units to the left and reflect across the -ais.. Use the function g=. Reflect the function across the -ais and move it down units.. Use the function g=. Move the function 7 units to the left, reflect across the -ais, and reflect across the -ais.. Use the function g=. Move the function 8 units to the right, units up, and reflect across the -ais.. f = + 5. f =. f = +7. f = 8 5. Determine if each of the following relations is a function. If so, determine whether it is even, odd, or neither. Also determine if it has -ais smmetr, -ais smmetr, origin smmetr, or none of the above, and then sketch the graph of the relation. See Eample f = +. g= 7. h = 8. w= 9. = 5. = 5. + = 5. F= ( ) 5. = + 5. = 55. g= 5 5. m = 5 Even function; -ais smmetr 57. = = Odd function; Origin smmetr Neither; No smmetr Odd function; Origin smmetr Not a function; -ais smmetr Neither; No smmetr Neither; No smmetr Not a function; -ais smmetr Even function; -ais smmetr 58. Odd function; Origin smmetr Not a function; -ais smmetr Neither; No smmetr Not a function; -ais smmetr Neither; No smmetr

68 . 7 Chapter Combining Functions TOPICS. Combining functions arithmeticall. Composing functions. Decomposing functions. Recursive graphics TOPIC Combining Functions Arithmeticall In Section.5, we gained eperience in building new functions from old ones b shifting, reflecting and stretching the old functions. In this section, we will eplore more was of building functions. We begin with four arithmetic was of combining two or more functions to obtain new functions. The basic operations are ver familiar to ou: addition, subtraction, multiplication, and division. The difference is that we are appling these operations to functions, but as we will see, the arithmetic combination of functions is based entirel on the arithmetic combination of numbers. DEFINITION Addition, Subtraction, Multiplication, and Division of Functions Let f and g be two functions. The sum f + g, difference f g, product f g, and quotient f are four new functions defined as follows: g f + g f g = + ( f g)= f g ( fg)= f g f ( )= g f g, provided that g ( ) The domain of each of these new functions consists of the common elements (or the intersection of elements) of the domains of f and g individuall, with the added condition that in the quotient function we have to omit those elements for which g =. With the above definition, we can determine the sum, difference, product, or quotient of two functions at one particular value for, or find a formula for these new functions based on the formulas for f and g, if the are available.

69 Combining Functions Section. 75 Combining Functions Arithmeticall EXAMPLE Given that f ( ) Solution: ( ) = 5 and g =, find f g and f g ( ). B the definition of the difference and quotient of functions, and f g = f ( ) g = 5 = 8, f f g ( )= g 5 = 5 =. EXAMPLE Combining Functions Arithmeticall Given the two functions f = Solution: and g=, find f + g and ( fg). B the definition of the sum and product of functions, ( f + g)= f + g and = + ( fg) = f g, ( ) = 5 =. What are the domains of f + g and f g? We first need to find the domains of the individual functions f and g. Domain of f: (, ) since f is a quadratic function Domain of g: [, ) since square roots of negative numbers are undefined Since the domain of two functions combined arithmeticall is the intersection of the individual domains, f + g and f g both have a domain of,. [ )

70 7 Chapter EXAMPLE Combining Functions Arithmeticall Given the graphs of f and g below, determine the domain of f + g and f g f ( f + g)() and g (). and evaluate f g Solution: From the graph, we can see that the domain of both f and g is the set of all real numbers (, ). This means that the domain of f + g is also (, ). To find the domain of the quotient, we need to check where g=. The graph shows us that this occurs when =±, so the domain of f is all real numbers ecept and : g (, ) (, ) (, ) To evaluate the new functions, we need to find f () and g() using the graph: We can see that f ()= and g()=, which means: f ( f + g)()= + = and g ()= =. TOPIC Composing Functions A fifth wa of combining functions is to form the composition of one function with another. Informall speaking, this means to appl one function to the output of another function. The smbol for composition is an open circle. DEFINITION Composing Functions Let f and g be two functions. The composition of f and g, denoted f g, is the function defined b ( f g)= f ( g ). The domain of f g consists of all in the domain of g for which g( ) is in turn in the domain of f. The function f g is read f composed with g, or f of g.

71 Combining Functions Section. 77 The diagram in Figure is a schematic of the composition of two functions. To calculate ( f g) we first appl the function g, calculating g, then appl the function f to the result, calculating f g f g. ( )= f g g g f f ( g ) Figure : Composition of f and g As with the four arithmetic was of combining functions, we can evaluate the composition of two functions at a single point, or find a formula for the composition if we have been given formulas for the individual functions. CAUTION! Note that the order of f and g is important. In general, we can epect the function f g to be different from the function g f. In formal terms, the composition of two functions, unlike the sum and product of two functions, is not commutative. EXAMPLE Composing Functions Given f = and g=, find: a. ( f g) c. f g b. g f d. ( g f) Solutions: = a. Since f g f g, the first step is to calculate g: g= = Then, appl f to the result: = ( )= = = f g f g f 9.

72 78 Chapter b. This time, we begin b finding f : f = = Now, appl g to the result: g( f ( ))= g( )= =. c. To find the formula for f g we appl the definition of composition, then simplif: ( f g)= f g = f ( ) = ( ) = + 9 Write out the definition of composition. Substitute the formula for g. Appl the formula for f. Simplif. d. To find a formula for the function g f we follow the same process: ( g f)= g f = g ( ) Write out the definition of composition. Substitute the formula for f. = Appl the formula for g ; the result is alread simplified. Note that once we have found formulas f g and g f parts b directl plugging into these formulas: we can answer the first two ( f g)= + 9 = 9 ( g f)= = CAUTION! When evaluating the composition f g value might be undefined: is not in the domain of g. Then g is not in the domain of f. Then f g g = In either case, f g f g at a point, there are two reasons the. is undefined and we can t evaluate f g is undefined and we can t evaluate it. is undefined, and is not in the domain of f g.

73 Combining Functions Section. 79 EXAMPLE 5 Domains of Compositions of Functions Let f = 5 and g=. Evaluate the following: + a. f g b. f g ( ) () Solutions: = a. f g f g ( ) But, if we tr to evaluate g, we see that it is undefined, so f g undefined. ()= () b. f g f g First, we evaluate g(). g ()= = + = We plug this result into f f g is also undefined. () but see that 5 is also = is undefined. Thus, EXAMPLE Domains of Compositions of Functions Let f = and g=. Find formulas and state the domains for: a. f g b. g f Solutions: = a. f g f g = f ( ) = Substitute the formula for g into f. Simplif. = While the domain of is the set of all real numbers, the domain of f g is, [ ) since onl nonnegative numbers can be plugged into g. ( )= b. g f g f = g = Substitute the formula for f into g. The answer is alread simplified. The domain of g f consists of all for which, or. We can write this in interval form as (, ],. [ )

74 8 Chapter TOPIC Decomposing Functions Often, functions can be best understood b recognizing them as a composition of two or more simpler functions. We have alread seen an instance of this: shifting, reflecting, stretching, and compressing can all be thought of as a composition of two or more functions. For eample, the function h =( ) is a composition of the functions f = and g= : f ( g )= f = = h. To decompose a function into a composition of simpler functions, it is usuall best to identif what the function does to its argument from the inside out. That is, identif the first thing that is done to the variable, then the second, and so on. Each action describes a less comple function, and can be identified as such. The composition of these functions, with the innermost function corresponding to the first action, the net innermost corresponding to the second action, and so on, is then equivalent to the original function. Decomposition can often be done in several different was. Consider, for eample, the function f = 5. Below we illustrate just a few of the was f can be written as a composition of functions. Be sure ou understand how each of the different compositions is equivalent to f.. g= h = 5. g= h = 5. g h j = = 5 = ( ) g h = g 5 = 5 = f ( ) g h = g 5 = 5 = f g( h( j )) = g h( ) = g 5 = 5 = f

75 Combining Functions Section. 8 EXAMPLE 7 Decomposing Functions Decompose the function f = + into: a. a composition of two functions b. a composition of three functions Solutions: a. g= + h = b. g h j = + = = g( h ) = g( ) = + = f g( h( j )) = g h( ) = g = + = f Note: These are not the onl possible solutions for the decompositions of f. TOPIC Recursive Graphics Recursion, in general, refers to using the output of a function as its input, and repeating the process a certain number of times. In other words, recursion refers to the composition of a function with itself, possibl man times. Recursion has man varied uses, one of which is a branch of mathematical art. There is some special notation to describe recursion. If f is a function, f this contet to stand for f ( f ), or ( f f), not f for f ( f ( f )), or f f f of f, with f n being the n th iterate of f. is used in! Similarl, f stands, and so on. The functions f, f, are called iterates Some of the most famous recursivel generated mathematical art is based on functions whose inputs and outputs are comple numbers. Recall from Section. that ever comple number can be epressed in the form a + bi, where a and b are real numbers and i is the imaginar unit. A one-dimensional coordinate sstem, such as the real number line, is insufficient to graph comple numbers, but comple numbers are easil graphed in a two-dimensional coordinate sstem. To graph the number a + bi, we treat it as the ordered pair ( a, b ) and plot it as a point in the Cartesian plane, where the horizontal ais represents pure real numbers and the vertical ais represents pure imaginar numbers. Benoit Mandelbrot used the function f ( z)= z + c, where both z and c are variables representing comple numbers, to generate the image known as the Mandelbrot set in the 97s. The basic idea is to evaluate the sequence of iterates f = + c = c,

76 8 Chapter. a. b. 8 c. 5 d. 5. a. b. c. d.. a. b. c. d.. a. b. c. d. 5. a. b. 8 c. 5 d. 5. a. b. 8 c. d. 7. a. b. c. d. 8. a. b. c. 7 d a. b. c. 9 d.. a. b. c. d.. a. 5 b. c. d.. a. b. c. d.. a. b. 5 c. d.. a. b. 5 c. d. 5. a. +, b. Dom = [, ), Dom = (, ) f = f ( c)= c + c, f f c c c = ( + )=( + c) + c, for various comple numbers c and determine if the sequence of comple numbers stas close to the origin or not. Those comple numbers c that result in so-called bounded sequences are colored black, while those that lead to unbounded sequences are colored white. The author has used similar ideas to generate his own recursive art, as described below. The image i of the storm reproduced here is based on the function ( iz ) + ( 7+ iz ) f ( z)=, where again z is a 5 z + variable that will be replaced with comple numbers. The image is actuall a picture of the comple plane, with the origin in the ver center of the golden ring. The golden ring consists of those comple numbers that lie a distance between.9 and. units from the origin. The rules for coloring other comple numbers in the plane are as follows: given an initial comple number z not on the gold ring, f ( z) is calculated. If the comple number f z colored the deepest shade of green. If not, the iterate f lies somewhere on the gold ring, the original number z is ( z) is calculated. If this result lies in the gold ring, the original z is colored a bluish shade of green. If not, the process continues up to the th iterate f ( z), using a different color each time. If f ( z) lies in the gold ring, z is colored red, and if not the process halts and z is colored black. The idea of recursion can be used to generate an number of similar images, with the end result usuall striking and often surprising even to the creator. Eercises ( ) In each of the following problems, use the information given to determine a. f + g f b. ( f g) ( ), c. ( fg) ( ), and d. g ( ). See Eamples,, and.. f ( )= and g( )= 5. f ( )= and g( )= = = = =. f and g. f and g f ( )= 5 and g( )=. f = and g= 7. f = + and g= + 8. f and g { ( ) ( )} = ( ) 9. f = ( 5, ),,,,,, and g,, 5, { ( )} =. f = ( 5, ),,,, and g = = { },

77 Combining Functions Section. 8. a. +, Dom = R b.,. f. g Dom = (, ), 7. a. +, Dom = R g f b. +, Dom = (, ) (, ) (, ) 8. a. +, [ ) Dom =, b., Dom =,, 9. a. + 8, Dom = R [ ) ( ) b. 8, Dom = (, ),. a. + +, Dom = [, ) b. +, Dom =,. a. + +, Dom = R b. +, Dom = (, ),. f g. g f In each of the following problems, find a. the formula and domain for f + g, and b. the formula and domain for f. See Eamples and. g 5. f = g= = = and. f and g = = = = 7. f and g 8. f and g = = = + = 9. f and g 8. f and g = =. f = and g = +. f and g. a. +, Dom = R b., Dom = (, ),.. π

78 8 Chapter. a., Dom = (, ], [ ) b., Dom = R 5. a., Dom = (, ), b., Dom = (, ) (, ). a., Dom = (, ) (, ) b., Dom =,, 7. a., Dom = [, ) b., Dom = (, ] 8. a., Dom = R b. +, Dom = R 9. a. +,. a. Dom = R b. +, Dom = R, [ ) Dom =, b. +, Dom =, [ ) In each of the following problems, use the information given to determine f g See Eamples and 5. = = = =. f 5 and g 5. f π π and g π = = = = 5. f and g. f 9 and g = + = + 7. f and g 8. f = and g = = + = 9. f and g.. g f f g. f. = =. and g f g In each of the following problems, find a. the formula and domain for f g, and b. the formula and domain for g f. See Eample.. f = and g= 5. f = and g=. f = 8. f = and g = + and g= 7. f = and g= = + = 9. f and g

79 Combining Functions Section. 85. a , Dom = R b. +, Dom = R. a., Dom = R b. 9 +, Dom = R + 7. a., Dom = R b , Dom = R. g=, h =, f = g h +. f = and g=. f = + and g= +. f = + and g= +. f = + and g = Write the following functions as a composition of two functions. Answers will var. See Eample 7.. f = 5. f = 5. f = + 7. f = f = f = 5. f = 5. f = + 5. f = In each of the following problems, use the information given to find g(). 5. f = + and f + g = f = and ( f g)= 55. f = and ( f g)= g=, h = 5, f = g h. g= +, h =, f = g( h ) 7. g= + 5, h = +, f = g h 8. g= +7, h = 5, f = g h = = f and g f 5 Solve the following application problems. 57. The volume of a right circular clinder is given b the formula V =πr h. If the height h is three times the radius r, show the volume V as a function of r. 58. The surface area S of a wind sock is given b the formula S = πr r + h, where r is the radius of the base of the wind sock and h is the height of the wind sock. As the wind sock is being knitted b an automated knitter, the height h increases with time t according to the formula ()= ht t. Find the surface area S of the wind sock as a function of time t and radius r. h r

80 8 Chapter 9. g=, h =, f = g( h ) 5. g=, h =, f = g( h ) 5. g=, h= +, = f g h 5. g=, h =, f = g h 5. g= g= 55. g= V = πr t 58. S = πr r V = πr t. ct ()= + 7, t. 5t =. f g f g = = ( f g) = 9, 9. fg, ( fg) ( )= = ( fg) 59. The volume V of the wind sock described in the previous question is given b the formula V = π r h where r is the radius of the wind sock and h is the height of the wind sock. If the height h increases with time t according to the formula ht ()= t, find the volume V of the wind sock as a function of time t and radius r.. A widget factor produces n widgets in t hours of a single da. The number of widgets the factor produces is given b the formula nt ()=, t 5t, t 9. The cost c in dollars of producing n widgets is given b the formula cn = + 7. n. Find the cost c as a function of time t.. Given two odd functions f and g, show that f g is also odd. Verif this fact with the particular functions f = g= and 9. Recall that a function is odd if f ( )= f for all in the domain of f.. Given two even functions f and g, show that the product is also even. Verif this fact with the particular functions f = and g=. Recall that a function is even f ( )= f for all in the domain of f. As mentioned in Topic, a given comple number c is said to be in the Mandelbrot set if, for the function f ( z)= z + c, the sequence of iterates f, f, f, stas close to the origin (which is the comple number + i ). It can be shown that if an single iterate falls more than units in distance (magnitude) from the origin, then the remaining iterates will grow larger and larger in magnitude. In practice, computer programs that generate the Mandelbrot set calculate the iterates up to a predecided point in the sequence, such as f 5, and if no iterate up to this point eceeds in magnitude, the number c is admitted to the set. The magnitude of a comple number a + bi is the distance between the point ( a, b ) and the origin, so the formula for the magnitude of a + bi is a + b. Use the above criterion to determine, without a calculator or computer, if the following comple numbers are in the Mandelbrot set or not.. c =. c = 5. c = i. c = 7. c = + i 8. c = i 9. c = i 7. c = i 7. c = 7. c =. Yes. No 5. Yes. Yes 7. No 8. Yes 9. No 7. No 7. No 7. Yes

81 Inverses of Functions Section.7 87 Inverses of Functions TOPICS.7.. Inverses of relations. Inverse functions and the horizontal line test Finding inverse function formulas TOPIC Inverses of Relations In man problems, undoing one or more mathematical operations plas a critical role in the solution process. For instance, to solve the equation + = 8, the first step is to undo the addition of on the left-hand side (b subtracting from both sides) and the second step is to undo the multiplication b (b dividing both sides b ). In the contet of more comple problems, the undoing process is often a matter of finding and appling the inverse of a function. We begin with the more general idea of the inverse of a relation. Recall that a relation is just a set of ordered pairs; the inverse of a given relation is the set of these ordered pairs with the first and second coordinates of each echanged. DEFINITION Inverse of a Relation Let R be a relation. The inverse of R, denoted R, is the relation defined b: { } R = ( b, a)( a, b) R. EXAMPLE Finding the Inverse of a Relation Determine the inverse of each of the following relations. Then graph each relation and its inverse, and determine the domain and range of both. { ( ) } = a. R =,,,, 5, b. Solutions: a. R =,,,, 5, R { ( ) } (, ),(, ),( 5, ) { } = For each ordered pair, switch the first and second coordinates (- and -coordinates). Recall that the domain is the set of first coordinates, and the range is the set of second coordinates.

82 88 Chapter R : R : { } { } { } Domain =,, Range = 5,, Domain = { 5,, } Range =,, 8 R In the graph to the left, R is in purple and its inverse is in green. The relation R consists of three ordered pairs, and its inverse is simpl these three ordered pairs with the coordinates echanged. Note that the domain of R is the range of R and vice versa. R { } { } b. R= (, ) = = = R, R : R : Domain = R Range = [, ) Domain =, Range = R R [ ) R In this problem, R is described b the given equation in and. The inverse relation is the set of ordered pairs in R with the coordinates echanged, so we can describe the inverse relation b just echanging and in the equation, as shown at left. Note that the shape of the graph of the relation and its inverse are essentiall the same. Consider the graphs of the two relations and their respective inverses in Eample. B definition, an ordered pair ( b, a ) lies on the graph of a relation R if and onl if ( a, b ) lies on the graph of R, so it shouldn t be surprising that the graphs of a relation and its inverse bear some resemblance to one another. Specificall, the are mirror images of one another with respect to the line =. If ou were to fold the Cartesian plane in half along the line = in the two eamples above, ou would see that the points in R and R coincide with one another. The two relations in Eample illustrate another important point. Note that in both cases, R is a function, as its graph passes the vertical line test. B the same criterion, R in Eample a is also a function, but R in Eample b is not. The conclusion to be drawn is that even if a relation is a function, its inverse ma or ma not be a function.

83 Inverses of Functions Section.7 89 TOPIC Inverse Functions and the Horizontal Line Test We have a convenient graphical test for determining when a relation is a function (the Vertical Line Test); we would like to have a similar test about determining when the inverse of a relation is a function. In practice, we will onl be concerned with the question of when the inverse of a function f, denoted f, is itself a function. CAUTION! We are faced with another eample of the reuse of notation. f does not stand for f when f is a function! We use an eponent of to indicate the reciprocal of a number or an algebraic epression, but when applied to a function or a relation it stands for the inverse relation. Assume that f is a function. f will onl be a function itself if its graph passes the vertical line test; that is, onl if each element of the domain of f is paired with eactl one element of the range of f. This is identical to saing that each element of the range of f is paired with eactl one element of the domain of f. In other words, ever horizontal line in the plane must intersect the graph of f no more than once. THEOREM The Horizontal Line Test Let f be a function. We sa that the graph of f passes the horizontal line test if ever horizontal line in the plane intersects the graph no more than once. If f passes the horizontal line test, then f is also a function. Of course, the horizontal line test is onl useful if the graph of f is available to stud. We can also phrase the above condition in a nongraphical manner. The inverse of f will onl be a function if for ever pair of distinct elements and in the domain of f, we have f f. This criterion is important enough to merit a name. DEFINITION One-to-One Functions A function f is one-to-one if for ever pair of distinct elements and in the domain of f, we have f ( ) f. This means that ever element of the range of f is paired with eactl one element of the domain of f. To sum up: the inverse f of a function f is also a function if and onl if f is one-to-one and f is one-to-one if and onl if its graph passes the horizontal line test.

84 9 Chapter EXAMPLE Inverse Functions Note: Even when a function f does not have an inverse function, it alwas has an inverse relation. Determine if the following functions have inverse functions. a. f = b. g= + Solutions: a. The function f does not have an inverse function, a fact demonstrated b showing that its graph does not pass the horizontal line test. We can also prove this algebraicall: although, we have f ( )= f ( ). Note that it onl takes two ordered pairs to show that f does not have an inverse function. b. The graph of g is the standard cubic shape shifted horizontall two units to the left. We can see this graph passes the horizontal line test, so g has an inverse function. Algebraicall, an two distinct elements of the domain of g lead to different values when plugged into g, so g is one-to-one and hence has an inverse function.

85 Inverses of Functions Section.7 9 Consider the function in Eample a again. As we noted, the function f = is not one-to-one, and so cannot have an inverse function. However, if we restrict the domain of f b specifing that the domain is the interval [, ), the new function, with this restricted domain, is one-to-one and has an inverse function. Of course, this restriction of domain changes the function; in this case the graph of the new function is the righthand half of the graph of the absolute value function. TOPIC Finding Inverse Function Formulas In appling the notion of the inverse of a function, we will often begin with a formula for f and want to find a formula for f. This will allow us, for instance, to transform equations of the form f = into the form = f. Before we discuss the general algorithm for finding a formula for f, consider the problem with which we began this section. If we define f = +, the equation + = 8 can be written as f = 8. Note that f is one-to-one, so f is a function. If we can find a formula for f, we can transform the equation into = f ( 8 ). This is a complicated wa to solve this equation, but it illustrates how to find inverses. What should the formula for f be? Consider what f does to its argument. The first action is to multipl b, and the second is to add. To undo f, we need to negate these two actions in reverse order: subtract and then divide the result b. So, f =. Appling this to the problem at hand, we obtain 8 = f = 8 =. This method of analzing a function f and then finding a formula for f b undoing the actions of f in reverse order is conceptuall important and works for simple functions. For other functions, however, the following algorithm ma be necessar as a standardized wa to find the inverse formula. PROCEDURE Formulas of Inverse Functions Let f be a one-to-one function, and assume that f is defined b a formula. To find a formula for f, perform the following steps: Step : Replace f in the definition of f with the variable. The result is an equation in and that is solved for at this point. Step : Interchange and in the equation. Step : Solve the new equation for. Step : Replace the in the resulting equation with f.

86 9 Chapter EXAMPLE Finding Formulas of Inverse Functions Find the inverse of each of the following functions. a. f = ( ) + Solutions: a. f = ( ) + = = = = + = + + f = + b. g= + Following the algorithm shows us how the steps of the original function get undone. First, replace f with. Net, switch and in the equation. To solve the resulting equation for, first subtract from both sides. Take the cube root of both sides. Add to both sides. Replace with f. b. g= + = + = + ( + )= + = = ( )= g = = The inverse of the function g is most easil found b the algorithm. The first step is to replace g with. The second step is to interchange and in the equation. We now have to solve the equation for. Begin b clearing the equation of fractions, and then proceed to collect all the terms that contain on one side. Factoring out the on the left-hand side and dividing b completes the process. The last step is to rename the formula g. Remember that the graphs of a relation and its inverse are mirror images of one another with respect to the line = ; this is still true if the relations are functions. We can demonstrate this fact b graphing the function and its inverse from Eample a above, as shown in Figure.

87 Inverses of Functions Section.7 9 f f Figure : Graph of a Function and Its Inverse We can use the functions and their inverses from Eample to illustrate one last important point. The ke characteristic of the inverse of a function is that it undoes the function. This means that if a function and its inverse are composed together, in either order, the resulting function has no effect on an allowable input! THEOREM Composition of Functions and Inverses Given a function f and its inverse f, the following statements are true. ( )= f f for all Dom f, and f ( f )= for all Dom( f ). For eample, given f = ( ) + and f = : f f ( )= f ( ) + + = ( ) + + = ( ) + = + =. A similar calculation shows that f ( f )=, as ou should verif. As another eample, consider g= + and g = :

88 9. Chapter... Dom = {,, } Ran = {,, } Dom = {,,, } Ran = {,,, } Dom = Ran = R g ( g )= g + = + + Similarl, g g ( )=, as ou should verif. = = + 7 = 7 = Eercises 5. [ ) Dom =, Ran = R Graph the inverse of each of the following relations, and state its domain and range. See Eample. { ( ) ( )} { ( ) }. R =,,,,,,,. S =,,,,,,,. =. = +. Dom = R [ ) Ran =, 5. =. = 7. = 8. = + = +. T = (, ), (, ), (, ), (, ) 9. { } 7. [ ) ( ] Dom =, Ran =,. =. = Dom = R Ran = R Dom = R Ran = R [ ) [ ) Dom =, Ran =, Dom = {,, } Ran = {,,, } Dom = R [ ) Ran =,

89 Inverses of Functions Section [ ) [ ) Dom =, Ran =, Dom = R Ran = R. Not a one-to-one function f ( 5) = f () = 5 5. Not a one-to-one function f ( ) = f () =. Dom = R Ran = R 7. Restrict to, 8. Inverse eists 9. Inverse eists [ ). Restrict to,. Inverse eists. Restrict to,. Inverse eists. Inverse eists [ ) 5. Restrict to,. Inverse eists [ ) [ ) [ ) 7. Restrict to, 8. Restrict to, 9. f = +. g + = Determine if each of the following functions is a one-to-one function. If so, graph the inverse of the function and state its domain and range.. 5. = + =.. = + = Determine if the following functions have inverse functions. If not, suggest a domain to restrict the function to so that it would have an inverse function (answers will var). See Eample. 7. f = + 8. g= ( ) 9. h = +. s= + 5. r=. b = 5. f =. m = 7. H= 8. p = Find a formula for the inverse of each of the following functions. See Eample. 9. f =. g=. r= +. s ( )= +. F= ( 5) +. G= V=. W= 5 7. h=

90 9 Chapter. r =. s = +. F = ( ) + 5. G + = 5. V = 5. W = 7. h = A ( 5 )= 9. J =. k = +. h = 7. F = A= J=. k= 7. h = +. F r=. P= ( + ) 5. f =. q= ( ) +, In each of the following problems, verif that f f ( )= and that f ( f )= f = and f = 8. f = + and f = ( + ) f = and f = 5. f =, and f = + 5. f = and f = 5. f = + and f = f = and f = 5. f = and f = f = ( ), and f = +, 5. f = f = and + Match the following functions with the graphs of the inverses of the functions. The graphs are labeled a. through f. 57. f= a. b. 5. r =. P = 5. f = f = f= c. d.. q =. f =, ( ) Answers will var. 57. b. f= e. f. 58. f 59. e. f= +. c

91 Inverses of Functions Section a. d FRISBEE VOLLEYBALL AND HORSESHOES. REMEMBER YOUR SUNBLOCK 7. CATCH A WAVE 8. BEACH FUN IN THE SUN An inverse function can be used to encode and decode words and sentences b assigning each letter of the alphabet a numerical value (A =, B =, C =,..., Z = ). Eample: Use the function f() = to encode the word PRECALCULUS. The encoded message would be The word can then be decoded b using the inverse function f =. The inverse values are which translates back to the word PRECALCULUS. Encode or decode the following words using the numerical values A =, B =, C =,..., Z =.. Encode the message SANDY SHOES using the function f () =.. Encode the message WILL IT RAIN TODAY using the function f = The following message was encoded using the function f () = 8 7. Decode the message The following message was encoded using the function f () = 5 +. Decode the message The following message was encoded using the function f=. Decode the message The following message was encoded using the function f() = 5. Decode the message

92 98 Chapter Chapter Project The Ozone Laer As time goes on, there is continuall increasing awareness, controvers, and legislation regarding the ozone laer and other environmental issues. The hole in the ozone laer over the south pole disappears and reappears annuall, and one model for its growth assumes the hole is circular and that its radius grows at a constant rate of. kilometers per hour.. Write the area of the circle as a function of the radius, r.. Assuming that t is measured in hours, that t = corresponds to the start of the annual growth of the hole, and that the radius of the hole is initiall, write the radius as a function of time, t.. Write the area of the circle as a function of time, t.. What is the radius after hours? 5. What is the radius after 5.5 hours? PHOTO COURTESY OF NASA. What is the area of the circle after hours? 7. What is the area of the circle after 5.5 hours? 8. What is the average rate of change of the area from hours to 5.5 hours? 9. What is the average rate of change of the area from 5.5 hours to 8 hours?. Is the average rate of change of the area increasing or decreasing as time passes?