1.2 Functions and Their Properties PreCalculus

Size: px

Start display at page:

Download "1.2 Functions and Their Properties PreCalculus"

Easter Carpenter
6 years ago
Views:

1. Functions and Their Properties PreCalculus 1. FUNCTIONS AND THEIR PROPERTIES Learning Targets for 1. 1. Determine whether a set of numbers or a graph is a function.

1 1. Functions and Their Properties PreCalculus 1. FUNCTIONS AND THEIR PROPERTIES Learning Targets for Determine whether a set of numbers or a graph is a function. Find the domain of a function given an set of numbers, an equation, or a graph 3. Describe the tpe of discontinuit in a graph as removable or non-removable 4. For a given function, describe the intervals of increasing and decreasing 5. Label a graph as bounded above, bounded below, bounded, or unbounded. 6. Find all relative etrema on a graph 7. Understand the difference between absolute and relative etrema 8. Describe the smmetr of a graph as odd or even This section is full of vocabular (obvious from the list above?). We will be investigating functions, and ou will need to answer questions to determine how each of these properties are applied to various functions. Function Definition and Notation In the last chapter, we used the phrase is a function of. But what is a function? In Algebra 1, we defined a function as a rule that assigned one and onl one (a unique) output for ever input. We called the input the domain and the output the range. Usuall, the set of possible -values is the domain, and the resulting set of possible -values is the range. Definition: Function A function from a set D to a set R is a rule that assigns a unique element in R to each element in D. To determine whether or not a graph is a function, ou can use the vertical line test. If an vertical line intersects a graph more than once, then that graph is NOT a function. Eample 1: A relation is ANY set of ordered pairs. State the domain and range of each relation, then tell whether or not the relation is a function. a) {( 3,0 ),( 4, ),(, 6) } b) {( 4, ),( 4, ),( 9, 3)( 9, 3) } c) d) Unit - 1

2 1. Functions and Their Properties PreCalculus For man functions, the domain is all real numbers, or. We tpicall start with this, and then see if there are an values of that cannot be used. The domain of a function can be restricted for 3 reasons that ou need to be aware of in this course: 1. NO negatives no negative numbers inside a square root. ** NO zeros no zeros in the denominator of a fraction 3. logno negatives... and NO zeros. no negatives and no zeros inside a logarithm Eample : Without using a graphing calculator, what is the domain of each of the following functions? 9 + a) = log b) h( ) = c) = Continuit In non technical terms, a function is continuous if ou can draw the function without ever lifting our pencil. Eample 3: Tell whether the following graphs are continuous or discontinuous. If discontinuous, label each tpe of discontinuit as removable or non-removable. Unit -

3 1. Functions and Their Properties PreCalculus Increasing/Decreasing Functions While there are technical definitions for increasing and decreasing, just remember to read the graph LEFT to RIGHT. Eample 4: Using the graph below, what intervals is the function increasing? Decreasing? Constant? Boundedness You need to understand the difference between the following terms: BOUNDED BELOW: BOUNDED ABOVE: BOUNDED: Local and Absolute Etrema Etrema is the plural form of one etreme value. Etrema is one word that includes maimums and minimums. Local (or Relative) Etrema are -values bigger (or smaller) than. Absolute Etrema are -values bigger (or smaller) than. Eample 5: Suppose the following function is defined on the interval [a, b]. Label f (a) through f (e) as relative or absolute etrema. a c d e b Unit - 3

4 1. Functions and Their Properties PreCalculus Smmetr The net topic we concern ourselves with when dealing with functions is the idea of smmetr. Smmetr on a graph means the functions look the same on one side as it does on another. We are most concerned with the tpes of smmetr that can be eplored numericall and algebraicall in terms of ODD and EVEN functions. 3 Tpes of Smmetr Smmetr with respect to the -ais: EVEN FUNCTIONS Graphicall Numericall Algebraicall Smmetr with respect to the origin: ODD FUNCTIONS Graphicall Numericall Algebraicall Smmetr with respect to the -ais: NOT A FUNCTION Graphicall Numericall Algebraicall Unit - 4

5 1. Functions and Their Properties PreCalculus 1. FUNCTIONS AND THEIR PROPERTIES Learning Targets for Prove a function is odd, even, or neither 10. Given a rational equation, find vertical asmptotes and holes (if the eist) 11. Given a rational equation, find the end behavior model 1. Given a rational equation, describe the end behavior using end behavior and limit notation. Odd vs Even Functions While there are man functions out there that are neither even nor odd, our concern with odd and even functions is twofold #1: Identif graphs of functions that are Odd or Even #: PROVE a function is Odd or Even While the first item above can be done graphicall, numericall, or algebraicall, the second is done ONLY algebraicall. Eample 6: Prove each function is odd, even, or neither. g = a) f ( ) = b) ( ) Vertical Asmptotes Eample 7: Graph the function f ( ) = What happens at = 3? Wh? Eample 8: Graph the function g( ) 9 =. What happens at = 3? Wh? + 3 Vertical asmptotes occur in rational functions when. If ou plug a point into our function and get, ou should look for a removable discontinuit a.k.a. a hole. Unit - 5

6 1. Functions and Their Properties PreCalculus End Behavior As ±, we sa the end behavior of the function is a description of what value f () approaches. For now, we will focus on finding end behavior using a graphical approach. Eample 9: Describe the end behavior of each graph shown below. We can write the end behavior in a more compact form using limit notation. There is a specific (a.k.a. correct) wa of writing limits that ou should learn. Eample 10: Rewrite the given end behavior for f () using limit notation. Sketch a possible graph for f (). a) As, f ( ) b) As, f ( ) As, f ( ) As, f ( ) 0 IF f () a number as ± meaning the end behavior of the function is a number, we sa the function has a horizontal asmptote (at that number) more on this in section.7. Unit - 6

7 1.3 Twelve Basic Functions PreCalculus 1.3 TWELVE BASIC FUNCTIONS Lesson Targets for Graph and Identif all 1 parent functions. Graph a piecewise function Parent Function #1: Linear Function (book refers to this as the identit function): Equation: Graph this function (label 5 points) Smmetr: Boundedness: Asmptotes: Discontinuities: Increasing/Decreasing: Domain: Range: Etrema: Parent Function #: Quadratic Function (book refers to this as the squaring function): Equation: Graph this function (label 5 points) Smmetr: Boundedness: Asmptotes: Discontinuities: Increasing/Decreasing: Domain: Range: Etrema: Unit - 7

8 1.3 Twelve Basic Functions PreCalculus Parent Function #3: Cubic Function (the book refers to this as the cubing function) : Equation: Graph this function (label 3 points) Smmetr: Boundedness: Asmptotes: Discontinuities: Increasing/Decreasing: Domain: Range: Etrema: Parent Function #4: Inverse Linear Function (book refers to this as the Reciprocal Function): Equation: Graph this function (label points, a H.A., and a V.A.) Smmetr: Boundedness: Asmptotes: Discontinuities: Increasing/Decreasing: Domain: Etrema: Range: Unit - 8

9 1.3 Twelve Basic Functions PreCalculus Parent Function #5: Square Root Function: Equation: Graph this function (label 3 points) Smmetr: Boundedness: Asmptotes: Discontinuities: Increasing/Decreasing: Domain: Range: Etrema: Parent Function #6: Eponential Function: (the book uses onl base e) We will use the equation f b, where b > 1 represents, and 0 < b < 1 represents. Graph this function (label points and a H.A.) Smmetr: Boundedness: Asmptotes: Discontinuities: Increasing/Decreasing: Domain: Range: Etrema: Unit - 9

10 1.3 Twelve Basic Functions PreCalculus Parent Function #7: Logarithm Function: (the book onl uses a natural logarithm) f We would like ou to use log b Graph this function (label points and a V.A.) Smmetr: Boundedness: Asmptotes: Discontinuities: Increasing/Decreasing: Domain: Etrema: Range: Parent Function #8: Absolute Value Function: Equation: Graph this function (label 5 points) Smmetr: Boundedness: Asmptotes: Discontinuities: Increasing/Decreasing: Domain: Range: Etrema: Unit - 10

11 1.3 Twelve Basic Functions PreCalculus Parent Function #9: Greatest Integer Function: Equation: Graph this function (label at least 6 points) Smmetr: Boundedness: Asmptotes: Discontinuities: Increasing/Decreasing: Domain: Range: Etrema: Parent Function #10: Logistic Function: Equation: Graph this function (label 1 point and H.A.) Smmetr: Boundedness: Asmptotes: Discontinuities: Increasing/Decreasing: Domain: Etrema: Range: Unit - 11

12 1.3 Twelve Basic Functions PreCalculus We will spend a large amount of time with the net two during second semester. For now, the parent function is enough. Parent Function #11: Sine Function: Equation: = sin() Graph this function Smmetr: Boundedness: Asmptotes: Discontinuities: Increasing/Decreasing: Domain: Range: Etrema: Parent Function #1: Cosine Function: Equation: = cos() Graph this function Smmetr: Boundedness: Asmptotes: Discontinuities: Increasing/Decreasing: Domain: Range: Etrema: Unit - 1

13 1.5 Graphical Transformations PreCalculus 1.5 GRAPHICAL TRANSFORMATIONS Learning Targets for Be able to graph transformations of parent functions. Be able to adjust the function when there is a horizontal stretch/shrink AND a left/right movement. You MUST be able to graph the parent functions from 1.3 in order to successfull transform them. A transformation is a stretch/shrink, a reflection, or simple movement of a parent function horizontall or verticall. The general rule Inside s opposite and Outside s same Eample 1: Suppose ou are given the function f (). If a, b, c, and d are real numbers, our transformed function is a f bc d The bo below summarizes our transformation rules: Inside Outside f c f d / f c f d f b a f f 1 b 1 a f / f f : Make sure ou before ou. Unit - 13

14 1.5 Graphical Transformations PreCalculus Eample : Perform the following transformations on the graph of f () below: a) f 3 b) f c) f 3 Eample 3: Graph the following functions b transforming their parent functions: **MOST OFTEN MISSED** a) f 3 6 b) g log 5 3 Unit - 14

15 4.4 and 4.5 Graphing Trigonometric Functions Pre-Calculus Learning Targets 1. Appl translations to the three parent trigonometric functions.. Given the attributes of a trigonometric function write the equation of a trig. function. 3. Given a graph, identif the attributes of the function and write its equation. Before we begin, let s review the Parent Sine and Cosine curves we learned about in both Algebra and in our Parent Functions Lesson 1.3 in this unit. WARM UP: Label the scale for graphs below to identif 9 ke points what do ou notice? Sine Function: f( ) = sin Cosine Function: f( ) = cos Here is some important terminolog we use with trigonometric graphs. Periodic Functions: An function that has a positive number p such that f( + p) = f( ) for ever value of. In other words an function whose graph repeats itself at equal intervals. Period: The smallest value of p. In other words the length of interval needed to complete one ccle of the graph. Amplitude: ½ (maimum minimum) In other words the height of the curve from the midline or -ais. What about Tangent? Tangent Function: f( ) = tan Period: Amplitude: Vertical Asmptotes: Unit 15

16 To review our translations A number added inside effects the horizontal direction, but it s backwards, and a number added outside effects the vertical direction. When we appl these transformations to the trig functions, we get = sin [ ( )] + or cos [ ( )] a b c d Amplitude: Period: Phase Shift: Vertical Shift: = a b c + d Eample 1: List the amplitude, period, phase shift and vertical shift. Then graph at least one period of the function. Be sure to label the scale of each ais. π a) = 4sin + b) = cos π c) = sin d) = tan 5 4 For this last eample, be careful the phase shift is NOT π! π e) = 3sin Unit 16

17 Eample : Write two equations of the cosine function whose amplitude is, period is 6π, phase shift is π 3 and vertical shift is 5. Eample 3: Write the equation for the given graph using a) the Sine Function 4 b) the Cosine Function. Unit 17

18 .3 Polnomial Functions of Higher Degree with Modeling PreCalculus.3 POLYNOMIAL FUNCTIONS OF HIGHER DEGREE WITH MODELING Learning Targets: 1. Be able to describe the end behavior of an polnomial using limit notation.. Understand how the multiplicit of a zero changes how the graph behaves when it meets the -ais. 3. Use end behavior and multiplicit of zeros to sketch a polnomial b hand. Our focus toda is on polnomials of degree 3 (cubic), degree 4 (quartic), or higher. End behavior describes what happens to the values of the function as approaches positive or negative infinit as we learned in lesson 1.. If ou can remember the graphs of =, =, =, and =, then ou can remember the end behavior of ALL polnomial functions using the leading coefficient and highest power. Even Power Odd Power Positive Coefficient lim f( ) = lim f( ) = lim f( ) = lim f( ) = Negative Coefficient lim f( ) = lim f( ) = lim f( ) = lim f( ) = Eample 1: Describe the end behavior of the polnomial function using lim f( ) and lim f( ). Confirm graphicall. 6 3 a) f ( ) = b) ( ) g = When the end behavior of a function goes to positive or negative infinit, we will write lim f( ) = or, but since infinit is not a real number, in Calculus we sa the limit does not eist. From Unit 1: The zeros of a function are the values that make f() = 0. These values are called the -intercepts when the are real numbers. We can use factoring to find these zeros algebraicall. The multiplicit of each zero is the number of times the factor occurs in the factored form of the polnomial. Unit - 18

19 .3 Polnomial Functions of Higher Degree with Modeling PreCalculus Eample : List each zero and state its multiplicit. Then, graph each function on our calculator. What do ou notice? a) f ( ) = ( 3) 3 b) h( ) = ( + 1)( ) ( 4) At a zero with EVEN multiplicit, the graph of the function will. At a zero with ODD multiplicit, the graph of the function will. Putting it all together Eample 3: State the degree and list the zeros of the polnomial function. State the multiplicit of each zero and whether the graph crosses the -ais at the corresponding -intercept. Using what ou know about end behavior and the zeros of the polnomial function, sketch the function. f = 5 a) ( ) 4 Degree: Sketch: Zero(s) & Multiplicit: b) ( ) ( )( ) h = Degree: Sketch: Zero(s) & Multiplicit: Unit - 19

asmptotes) DAY 1 iii) -intercept(s) DAY iv) -intercept DAY. Graph a rational function b hand including all of the above attributes (if the eist).

20 .7 Graphs of Rational Functions PreCalculus.7 Graphs of Rational Functions Da 1 Learning Targets: 1. Find the following attributes of rational functions: i) End Behavior (including Horizontal Asmptotes or Slant Asmptotes) DAY 1 ii) Vertical Asmptotes (distinguish between holes and vertical asmptotes) DAY 1 iii) -intercept(s) DAY iv) -intercept DAY. Graph a rational function b hand including all of the above attributes (if the eist). DAY All of the functions we will deal with for the remainder of this chapter will be Rational Functions. Let s begin with a definition and a review of end behavior f ( ) Rational Function: = g( ) Eample 1: Find the end behavior of the function using limit notation. NOTE: For ALL rational functions, IF the end behavior is a numeric value, that value will be the SAME as ±. In addition, the end behavior for all rational functions can fit into three categories. Eamples are provided below. A) f ( ) = 5+ 4 B) g( ) 3 4 = 7 3 C) h( ) 4 = + 5 Eample : Find the end behavior for each rational function above. a) b) c) Eample 3: Since end behavior onl uses the 1 st term of a polnomial and a rational function is simpl the ratio of two polnomials, we can use the ratio of the end behaviors to find the end behavior of the rational function. How does the ratio of the end behaviors compare to the end behavior of the graphs? Unit - 0

21 .7 Graphs of Rational Functions PreCalculus Here s wh this works 1 Step 1: Consider the parent function h( ) =. As, where does h () approach? Step : Let s rewrite g( ) 3 4 = 7 3 We need to put all this information together with the what we alread know about Domain, Vertical Asmptotes and Holes from previous chapters ep, ou were supposed to remember that!! Eample 4: Use the equation below to answer the following questions: 18 h 1115 a) Find the end behavior. Include HA or slant asmptotes. b) Find an vertical asmptotes. Distinguish between holes and vertical asmptotes. Eample 5: Use the equation below to answer the following questions: 68 g 1 a) Find the end behavior. Include HA or slant asmptotes. b) Find an vertical asmptotes. Distinguish between holes and vertical asmptotes. Unit - 1

22 .7 Graphs of Rational Functions PreCalculus.7 Graphs of Rational Functions Da Learning Targets: 1. Find the following attributes of rational functions: iii) -intercept(s) DAY iv) -intercept DAY. Graph a rational function b hand including all of the above attributes (if the eist). DAY An -intercept is an -value whose -value =. We will write our answer as. A -intercept is an -value whose -value =. We will write our answer as. Eample 1: Find the -intercept and the -intercept of f ( ) = 4 7. Let s see how these definitions appl to Rational Functions When does a fraction = 0? How do we find the -value of a fraction when =0? Eample : Find the -intercept and the -intercept of f ( ) = Now to put this information with what we learned about asmptotes in -7 da 1. We will revisit the same eamples. Eample 3: Given 18 ( 3)( 3) h 1115 (5)( 3) lim h ( ) = 1 a) Find the end behavior. HA: = 1 ± b) Find an vertical asmptotes. Hole at = 3; VA: = 5 c) Find the -intercept(s). d) Find the -intercept. e) Graph the function using the information above and an additional points as needed. Unit -

23 .7 Graphs of Rational Functions PreCalculus For eamples 4-5, for the given function find the following (if the eist). a. End Behavior including the equations of horizontal or slant asmptotes b. Vertical Asmptote(s). Distinguish between holes and vertical asmptotes. c. - intercept(s) d. -intercept and additional points in each region to determine the shape. e. Graph without a calculator. Eample 4: g 6 8 ( 4)( ) 1 1 a) Find the end behavior. SA: = + 5 b) Find an vertical asmptotes. VA: = 1 lim g ( ) = lim g ( ) = 3 1 k 3 Eample 5: Unit - 3

24 1.4 Building Functions from Functions PreCalculus 1.4 BUILDING FUNCTIONS FROM FUNCTIONS Learning Targets 1. Be able to Add, Subtract, Multipl, and Divide two functions. Be able to Compose two functions 3. Find an Inverse Functions Graphicall, Numericall, and Algebraicall 4. Prove (or Verif) two functions are Inverses Creating functions from other functions can be done in a variet of was. We are going to add, subtract, multipl, divide, and compose functions to create new functions. Eample 1: Epress in function notation what the following epressions mean: f g a) ( f + g)( ) b) ( f g)( ) c) ( fg )( ) d) ( ) Eample : Consider the two functions f ( ) = 6 and g( ) a) Find ( g f )( ). State the Domain of the new function. = +. What is the Domain of each? f g b) Find ( ). State the Domain of the new function. Composite Functions When the range of one function is used as the domain of a second function we call the entire function a composite function. We use the notation f g f g to describe composite functions. This is read as "f composed with g" or "f of g of ". The function f in the eample above is an eample of a composite function. The linear function + is applied first, then the square root function. Eample 3: Suppose f 1 and g. a) Find g f. What is the domain of g f? b) Find f g. What is the domain of f g? Unit - 4

1.4 Building Functions from Functions PreCalculus Inverse Functions A function has an inverse function if and onl if the original function passes the Horizontal Line Test.

25 1.4 Building Functions from Functions PreCalculus Inverse Functions A function has an inverse function if and onl if the original function passes the Horizontal Line Test. The Horizontal Line Test works just like the Vertical Line Test (it s just horizontal ). All an inverse function does is switch the and or the domain and range. Eample 4: The graphs of two functions are shown below. Do the have an inverse? Wh or wh not? h 3 1 k 5 Once we know whether a function has an inverse, our net task is to find an equation and/or a graph for the inverse. Finding the Inverse Graphicall: Reflect the graph of the original function over the line =. Finding the Inverse Numericall: Plot the reverse of the coordinates. Finding the Inverse Algebraicall ( THIS is our focus): Switch the and in the original equation, then solve the new 1 equation for in order to write as a function of. Your NEW can be written as f Eample 5: Let f 3 1, find the inverse algebraicall. Verifing Inverses It is one thing to find the inverse function (either graphicall or algebraicall), but it is another to verif that two functions are actuall inverses. Whenever ou are verifing anthing in mathematics, ou must go back and use the definition. Definition: Inverse Function A function f has an inverse f 1 if and onl if 1 f f f 1 f Eample 6: According to this definition, how man composite functions must be used to verif inverses? Eample 7: Verif that the function ou found in eample 5 is in fact the inverse function. Unit - 5

26 1.4 Building Functions from Functions PreCalculus Eample 8: Find f 1 and verif if 3 f. YEAH ou ll probabl need most of this space. Unit - 6

A function from a set D to a set R is a rule that assigns a unique element in R to each element in D.

A function from a set D to a set R is a rule that assigns a unique element in R to each element in D. 1.2 Functions and Their Properties PreCalculus 1.2 FUNCTIONS AND THEIR PROPERTIES Learning Targets for 1.2 1. Determine whether a set of numbers or a graph is a function 2. Find the domain of a function