Chapter 4: Exponential and Logarithmic Equations Section 4 ...· Chapter 4: Exponential and Logarithmic

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  • Section 4.1 Composite Functions 119

    Copyright 2015 Pearson Education, Inc.

    Chapter 4: Exponential and Logarithmic Equations Section 4.1: Composite Functions

    Exploration 1*: Form a Composite Function

    1. Suppose you have a job that pays $10 per hour. Write a function, g that can be used to determine your gross pay (your pay before taxes are taken out) per hour, h, that you worked.

    ( )g h

    2. Now lets write a formula for how much money youll actually take home of that paycheck. Lets assume your employer withholds 20% of your gross pay for taxes. Write a function, n, that determines your net pay based off of your gross income, g.

    ( )n g

    3. How much money would you net if you worked for 20 hours?

    4. Instead of having to use two different functions to find out your net pay, as you most likely did in (3), lets combine our functions from (1) and (2) and write them as one function. This is called composing functions. Write a function that relates the number of hours worked, h, to your net pay, n.

    Definition: Given two functions f and g, the composite function, denoted by __________ (read as f composed with g) is defined by________________________.

    Note: f g does not mean f multiplied by g(x). It means input the function g into the function f: ( ) ( )f g f g x f x g x The domain of f g is the set of all numbers x in the domain of g such that ( )g x is in the domain of f. In other words, f g is defined whenever both ( )g x and f g x are defined.

  • 120 Chapter 4 Exponential and Logarithmic Equations

    Copyright 2015 Pearson Education, Inc.

    Example 1*: Form a Composite Function; Evaluate a Composite Function Suppose that 22 3f x x and 34 1g x x . Find:(a) ( )(1) (b) ( )(1) (c) ( )( 2) (d) ( )( 1) f g g f f f g g Example 2: Evaluate a Composite Function If f (x) and g(x) are polynomial functions, use the table of values for f (x) and g(x) to complete the table of values for ( )( )f g x .

    Example 3: Find the Domain of a Composite Function Suppose that 22 3f x x and 34 1g x x . Find the following and their domains: (a) f g (b) g f

    x g x -2 4 -1 1 0 0 1 1 2 4

    x f x 0 3 1 4 2 5 3 6 4 7

    x ( )( )f g x -2 -1 0 1 3

  • Section 4.1 Composite Functions 121

    Copyright 2015 Pearson Education, Inc.

    Example 4*: Find the Domain of a Composite Function

    Find the domain of ( )( )f g x if 1 4 and 4 2

    f x g xx x

    .

    Example 5: Find a Composite Function and Its Domain

    Suppose that 1f xx

    and 1g x x . Find the following and their domains: (a) f g (b) f f

    Example 6*: Showing Two Composite Functions Are Equal

    If 12 and ,2

    f x x g x x show that f g x g f x x for every x in the domain of and .f g g f

  • 122 Chapter 4 Exponential and Logarithmic Equations

    Copyright 2015 Pearson Education, Inc.

    Example 7: Showing Two Composite Functions Are Equal

    If 1 1 and 2 1,2

    f x x g x x show that f g x g f x x for every x in the domain of and .f g g f Example 8*: Find the Components of a Composite Function Find functions f and g such that 4if 2 3f g H H x x Example 9: Find the Components of a Composite Function

    Find functions f and g such that 21 if

    2 3f g H H x

    x

  • Section 4.2 One to One Functions; Inverse Functions 123

    Copyright 2015 Pearson Education, Inc.

    Chapter 4: Exponential and Logarithmic Functions Section 4.2: One-to-One Functions; Inverse Functions

    Definition: A function is one to one if any two different inputs in the domain correspond to _________________________________________________. That is, if 1x and 2x are two different inputs of a function f, is one to one if __________________.

    Example 1*: Determine Whether a Function is One to One Determine whether the following functions are one to one. Explain why or why not.

    (a) (b) {(1,5), (2,8), (3,11), (4,14)}

    The Horizontal Line Test Theorem: If every horizontal line intersects the graph of a function f in at most ________________, then f is one to one.

    Why does this test work? You may want to refer to the definition of one to one functions.

    Dan

    John

    Joe

    Andy

    Saturn

    Pontiac

    Honda

    Student Car

  • 124 Chapter 4 Exponential and Logarithmic Equations

    Copyright 2015 Pearson Education, Inc.

    Example 2: Determine whether a Function is One to One Using the Horizontal Line Test For each function, use the graph to determine whether the function is one to one.

    Theorem: A function that is increasing on an interval I is a one to one function on I. A function that is decreasing on an interval I is a one to one function on I.

    Why is this theorem true?

    Exploration 1: Inverse Functions Reverse the Process You might have experienced converting between degrees Fahrenheit and degrees Celsius when measuring a temperature. The standard formula for determining temperature in degrees

    Fahrenheit, when given the temperature in degrees Celsius, is 9 325

    F C . We can use this

    formula to define a function named g, namely 9 325

    F g C C , where C is the number

    of degrees Celsius and g C is a number of degrees Fahrenheit. The function g defines a process for converting degrees Celsius to degrees Fahrenheit. 1. What is the value of 100g ? What does it represent?

    2. Solve the equation 112g C and describe the meaning of your answer.

    3. What happens if you want to input degree Fahrenheit and output degree Celsius?

    Reverse the process of the formula 9 325

    F C by solving for C.

  • Section 4.2 One to One Functions; Inverse Functions 125

    Copyright 2015 Pearson Education, Inc.

    Definition: Suppose that f is a one to one function. Then, to each x in the domain of f, there is _______________________ y in the range (because f is a function); and to each y in the range of f there is exactly one x in the domain (because f is one to one). The correspondence from the range of f back to the ______________ of f is called the inverse

    function of f. We use the symbol 1f to denote the inverse of f. Note: 1 1ff

    In other words, two functions are said to be inverses of each other if they are the reverse process of each other. Notice in the exploration, the formula found in part (c) was the reverse process of g. Instead of inputting Celsius and outputting Fahrenheit, the new function inputs Fahrenheit and outputs Celsius.

    Example 3*: Determine the Inverse of a Function Find the inverse of the following function. Let the domain of the function represent certain students, and let the range represent the make of that students car. State the domain and the range of the inverse function. Example 4*: Determine the Inverse of a Function Find the inverse of the following one to one function. Then state the domain and range of the function and its inverse.

    {(1,5), (2,8), (3,11), (4,14)} Domain and Range of Inverse Functions: Since the inverse function, 1f , is a reverse mapping of the function f : Domain of f = _____________ of 1f and Range of f = _______________ of 1f

    Dan

    John

    Joe

    Michelle

    Saturn

    Pontiac

    Honda

    Chrysler

    Student Car

  • 126 Chapter 4 Exponential and Logarithmic Equations

    Copyright 2015 Pearson Education, Inc.

    Fact: What f does, 1f undoes and vice versa. Therefore,

    1 ____f f x where x is in the domain of f 1 ____f f x where x is in the domain of 1f

    We can use this fact to verify if two functions are inverses of each other. Example 5*: Determine the Inverse of a Function ; Verifying Inverse Functions Verify that the inverse of 3 1 32 is 2g x x g x x by showing that 1g g x x for all x in the domain of g and that 1 g g x x for all x in the domain of -1.g Exploration 2: Graphs of Inverse Functions

    1. Using a graphing utility, graph the following functions on the same screen 3 3, , and y x y x y x

    2. What do you notice about the graphs of 3 3, its inverse ,y x y x and the line y x ?

    3. Repeat this experiment by graphing the following functions on the same screen: 1, 2 3, and ( 3)2

    y x y x y x

    4. What do you notice about the graphs of 12 3, its inverse ( 3),2

    y x y x and the

    line y x ?

  • Section 4.2 One to One Functions; Inverse Functions 127

    Copyright 2015 Pearson Education, Inc.

    Theorem: The graph of a one to one function f and the graph of its inverse 1f are symmetric with respect to the line ______________.

    Example 6*: Obtain the Graph of the Inverse Function The graph shown is that of a one to one function. Draw the graph of its inverse.

    Procedure for Finding the Inverse of a One to One Function Step 1: In ( )y f x , interchange the variables x and y to obtain ____________. This

    equation defines the inverse function 1f implicitly. Step 2: If possible, solve the implicit equation for y in terms of x to obtain the explicit form

    of 1f : ____________________. Step 3: Check the result by showing that _______________ and ________________. Example 7: Find the Inverse Function from an Equation

    Find the inverse of 1( ) 13

    f x x