Functions Some Basic Ideas and Some Examples

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1 Functions Some Basic Ideas and Some Examples S. F. Ellermeyer May 30, Basic Ideas and Terminology Let D < with D 6= A real valued function, f, ond is an assignment of each number x D toasinglerealnumber,y. If the function f assigns the number x to the number y, then we write f (x) =y whichisreadas f of x equals y. y is said to be the value of the function f at the point x. ThesetD is called the domain of f. Therange of f is the set R f = {y y = f (x) forsomex D}. 2 Examples of Functions 2.1 Example 1 Let D = { 5, 2, 0, 5, 17} and let f be the function on D which makes the following assignments: f ( 5) = 7 f ( 2) = 3 f (0) = 14 f (5) = 14 f (17) = 2 Note that the range or f is R f = { 3, 2, 7, 14}. 1

2 2.2 Example 2 Let D = {1, 2, 3, 4,...} (the set of all positive integers). Let f be the function on D defined by f (n) =gcd(n, 16) where gcd stands for greatest common divisor. Some values of f are shown below. f (1) = gcd (1, 16) = Exercises f (2) = gcd (2, 16) = 2 f (3) = gcd (3, 16) = 1 f (4) = gcd (4, 16) = 4 f (5) = gcd (5, 16) = 1 f (6) = gcd (6, 16) = 2 1. Compute f (7), f (8), f (9), f (416), and f (4023). 2. What is the range of f? 2.3 Example 3 Let D =( 5, 2) and let f be the function on D which is defined by the rule f (x) =3x 2 4x. Then, for example, f ( 4) = 3 ( 4) 2 4( 4) = 64 and f f (0) = 3 (0) 2 4(0)=0 µ 2 2 µ 2 =3 4 = µ 2 7 f (1.57) = 3 (1.57) 2 4(1.57) =

3 2.3.1 Exercise Compute f ( 2), f ( 1.1), f ³ 1, 2 andf (0.83). 2.4 Example 4 Let D =(, ) andletg be the function on D which is defined by the rule g (x) =3x 2 4x Exercises 1. Find g ( 4), g (1.57), and g (6). 2. Why is the function g different from the function f of the previous example? (Hint: In the previous example, is there such a thing as f (6)?) 2.5 Example 5 Let D =[ 2, 4] and let f be the rule defined by f (x) = x. Then f is not a function on D because f cannot be used to assign each point x D to a real number. For example, we cannot evaluate f ( 1) because 1isnotdefined (as a real number). 2.6 Example 6 Let D = < and let f be the function defined on D by ( 3x if x 2 f (x) = x +10 if x>2 Then, for example, f ( 4) = 3 ( 4) = 12 (because 4 2) f (0) = 3 (0) = 0 (because 0 2) 3

4 f (3) = = 13 (because 3 > 2) and f (4.97) = = (because 4.97 > 2). This type of function is called a piecewise defined function because different formulas define f on different parts of its domain Exercise Compute f ( 8), f (1.97), f (2), f (2.004), and f (10). 3 Some Remarks About Domains of Functions Defining a function involves specifying the domain on which the function is to act. For example, the function f defined by the rule f (x) =3x 2 4x on the domain D =( 2, 5) and the function g defined by the rule g (x) =3x 2 4x on the domain D =(0, )aredifferent functions even though the rules which define both functions are the same. They are different because their domains are different. In applications, the domain of a function involved in studying the application is often naturally prescribed. For example, suppose that the cost of a certain type of rope is 25 cents per foot and suppose we want to define a function which gives the cost (in dollars) of buying any given length of rope. Since the cost of each foot of rope is.25 dollars, then the cost of x feet of rope is.25x. The function which gives the cost of buying x feet of rope is C (x) =.25x. What is a natural choice of domain for the function C? Obviously, the domain of C should not include negative numbers because there is no such thing as a negative length of rope. An appropriate choice for the domain of C would be D =[0, ). In some problems involving functions, the domain of the function is not really an important issue. Indeed, you will often encounter statements like 4

5 Consider the function f (x) =3x 2 4x... with no reference made to the domain of the function. In such cases, the convention is to assume that the domain is the largest subset of < on which the rule for the function is defined. For example, if you encounter the statement Consider the function f (x) =3x 2 4x with no reference made to the domain, then you may assume that the (implied) domain is the set of all real numbers because the formula which defines f is defined for all x <. As another example, suppose that you are asked to work with the function f (x) = x + 1 x 2 without being told what the intended domain of this function is. The formula which defines f is not defined for any negative number because of the radical and the formula is also not defined for x = 2 because setting x =2would place a 0 in the denominator if the second term in the formula. However, the formula is defined for all other values of x so you could conclude that the intended domain of the function f is D =[0, 2) (2, ). (Note that another way to write this domain is D =[0, ) {2}.) In summary, if the domain of a function is not prescribed beforehand by the person who defines the function, then the domain can be deduced according to the following conventions: Convention 1. If the function arises in an application (a real problem), then the domain should be chosen so that it makes sense in terms of the problem. Convention 2. If there is no natural choice of domain (as in cases where Convention 1 applies), then the domain is assumed to be the largest subset of < on which the rule for the function is defined. 3.1 Exercises In exercises 1 5, a rule for a function is given but no domain is specified. Determine the implied domain of each function (via Convention 2). 1. f (x) =14x 5x g (u) =1/ (u +14) 5

6 3. h (u) =1/ (u 2 +14) 4. A (x) = x 4 5. p (t) = t/ ³ 9 t A farmer has 600 feet of fence available which he plans to use to enclose a rectangular area. Let x stand for the length of fence to be used on one side of the rectangular area and let A (x) be the area enclosed by the fence which is built. Write down therule(formula)forthefunction A and determine the domain of A. 7. A certain movie theater has a capacity of 800 people and charges $6 perticketformovies. Define a function R which gives the revenue collected by the movie theater at each movie showing as a function of the number of people in attendance. State the domain of R. 6

7 4 Describing Functions via Formulas, Tables, and Graphs Functions can be described using formulas, tables, and/or graphs. Depending on what you are trying to do with the function you are studying, any one of these methods for describing the function might come in handy. Let us examine each of the functions in Examples 1 6 above and describe each of them in terms of a formula, a table, and or a graph (whichever are possible). 4.1 Example 1 Let D = { 5, 2, 0, 5, 17} and let f be the function on D which makes the following assignments: f ( 5) = 7 f ( 2) = 3 f (0) = 14 f (5) = 14 f (17) = Formula It is not practical or necessary to try to describe this function using a formula. After all, there are only five points in the domain so the description of the function given above is perfectly adequate. Nonetheless, here is a formula for f: f (x) =14+ 3, 246, 658 1, 529, , , 241 x 373, 065 2, 487, 100 x2 746, 130 x3 + 2, 487, 100 x4. If you have a rainy Saturday afternoon with nothing better to do, you can check to see that this formula actually does correspond to the function f. It is easy to see, for example, that the formula gives f (0) = 14. You could use your calculator to check that the other x values in the domain of f give the indicated y values. Let us stress again though that it is really unnecessary (and, in fact, inconvenient) to describe f by a formula. 7

8 4.1.2 Table Since the function f acts on a domain containing only five points, f is most easily described by a table. The table for f is Graph x y The graph of f consists of just five points. This graph is pictured below (with small squares indicating the (x, y) pairs of f) x Example 2 Let D = {1, 2, 3, 4,...} (the set of all positive integers). Let f be the function on D defined by f (n) =gcd(n, 16) where gcd stands for greatest common divisor. 8

9 4.2.1 Formula This function is defined by the formula f (n) =gcd(n, 16) Table It would be impossible to make a complete table for this function because its domain contains infinitely many points. Of course, one could make a partial table as follows: Graph n y The graph of f consists of infinitely many scattered points. A graph is really not the best way to describe this function. Nonetheless, one could make a partial graph as pictured below x 4.3 Example 3 Let D =( 5, 2) and let f be the function on D which is defined by the rule f (x) =3x 2 4x Formula This function is defined by the formula f (x) =3x 2 4x. 9

10 4.3.2 Table The domain of f is an entire interval. Hence, it is not possible to produce a complete table listing all values of f. However, we could make a partial table which is as extensive as we like. Here is such a partial table (using x values chosen at a distance of 0.5 from each other): x y Graph Since the domain of f is an interval and since the values of f vary in a continuous fashion as x is varied, the graph of f is a curve in the Cartesian plane. Note that this was not the case for the graphs of the previous two functions in Examples 1 and 2. The graph of f is pictured below Example x Graph of f (x) =3x 2 4x with domain ( 5, 2) Let D =(, ) andletg be the function on D which is defined by the rule g (x) =3x 2 4x Formula The function g is defined by the formula g (x) =3x 2 4x. 10

11 4.4.2 Table As in the previous example, the domain of g contains infinitely many points so a complete table cannot be produced. A partial table can be produced if desired Graph The graph of g is shown below. Notice that, of course, this graph coincides with the graph of the function f from the previous example on the interval ( 5, 2) because f and g are both defined by the same formula. However, the graph of g extends infinitely both to the left and to the right. This is indicated by the arrows on each side of the graph Example x 4 Graph of g (x) =3x 2 4x with domain (, ) Recall that Example 5 was not even a function! 4.6 Example 6 Let D = < and let f be the function defined on D by f (x) = ( 3x if x 2 x +10 if x>2 11

12 4.6.1 Formula The function f is defined by the formula Table f (x) = ( 3x if x 2 x +10 if x>2 The domain of f is the set of all real numbers (an infinite set) so it is not possible to produce a complete table for f. Here is a partial table for f Graph The graph of f is pictured below. x y

13 5 Exercises 1. Let f be the function whose graph is shown below. (The complete graph of f is shown. It contains only six points.) Make a table for f and state the domain and the range of f x A partial table for a function g which has domain D =(, ) is shown below. Find a possible formula for g andsketchthegraphof g. (There is more than one possible solution to this exercise but one solution is easier than any other.) t v Let f bethefunctionwithdomaind =[ 12, 4] defined by f (x) = 4x 4. Sketch the graph of f and determine the range of f. 4. Let g be the function with domain D =(, ) defined by g (x) = 4x 4. Sketch the graph of g and determine the range of g. 5. Let f be defined by the rule f (x) = 1 x

14 a. What is the (implied) domain of f? b. Complete the following table for f. x y c. Use graph paper and the table you completed in part b to sketch an accurate graph of f. (If you aren t sure what the graph of f looks like, you may want to make a more extensive table than the one in part b.) d. What is the range of f? 6. Repeat Exercise 5 using f (x) = 1 x (For part b, its up to you to decide which x values to choose in making a table.) 7. Repeat Exercise 5 using f (x) = x 4 (For part b, its up to you to decide which x values to choose in making a table.) 8. Repeat Exercise 5 using f (x) = x 9 x +8. (For part b, its up to you to decide which x values to choose in making a table.) 14

WORKSHEET ON NUMBERS, MATH 215 FALL. We start our study of numbers with the integers: N = {1, 2, 3,...}

WORKSHEET ON NUMBERS, MATH 215 FALL 18(WHYTE) We start our study of numbers with the integers: Z = {..., 2, 1, 0, 1, 2, 3,... } and their subset of natural numbers: N = {1, 2, 3,...} For now we will not