SECTION 5.2 One-to-One Functions; Inverse Functions 293 (a) What is the domain of the function T? (b) Given that the tax due T is an increasing linear function of modified adjusted gross income g, find the range of the function T. (c) Find adjusted gross income g as a function of federal income tax T. What are the domain and the range of this function? 102. Income Taxes The function ( ) = + − T g g ( ) 2320 0.12 23,200 represents the 2024 federal income tax T (in dollars) due for a couple whose filing status is “married filing jointly” and whose modified adjusted gross income is g dollars, where < ≤ g 23,200 94,300. (a) What is the domain of the function T? (b) Given that the tax due T is an increasing linear function of modified adjusted gross income g, find the range of the function T. (c) Find adjusted gross income g as a function of federal income tax T. What are the domain and the range of this function? 103. Gravity on Earth If a rock falls from a height of 100 meters above Earth, the height H (in meters) after t seconds is approximately ( ) = − H t t 100 4.9 2 (a) In general, quadratic functions are not one-to-one. However, the function H is one-to-one. Why? (b) Find the inverse of H and verify your result. (c) How long will it take a rock to fall 80 meters? 104. Period of a Pendulum The period T (in seconds) of a simple pendulum as a function of its length l (in feet) is given by π ( ) = T l l 2 32.2 (a) Express the length l as a function of the period T. (b) How long is a pendulum whose period is 3 seconds? 105. Challenge Problem Given ( ) = + + f x ax b cx d find ( ) −f x . 1 If ≠ c 0, under what conditions on a, b, c, and d is f f ?1 = − 106. Challenge Problem If ( ) ( ) = h x f g x ( ) , find −h 1 in terms of −f 1 and −g .1 107. Challenge Problem For f x x x x x 2 3, 0 3 4, 0 { ( ) = + < + ≥ (a) Find the domain and range of f. (b) Find −f .1 (c) Find the domain and range of −f .1 In applications, the symbols used for the independent and dependent variables are often based on common usage. So, rather than using ( ) = y f x to represent a function, an applied problem might use ( ) = C C q to represent the cost C of manufacturing q units of a good. Because of this, the inverse notation −f 1 used in a pure mathematics problem is not used when finding inverses of applied problems. Rather, the inverse of a function such as ( ) = C C q will be ( ) = q q C . So ( ) = C C q is a function that represents the cost C as a function of the number q of units manufactured, and ( ) = q q C is a function that represents the number q as a function of the cost C. Problems 97–104 illustrate this idea. 97. Vehicle Stopping Distance Taking into account reaction time, the distance d (in feet) that a car requires to come to a complete stop while traveling r miles per hour is given by the function ( ) = − d r r 6.97 90.39 (a) Express the speed r at which the car is traveling as a function of the distance d required to come to a complete stop. (b) Verify that ( ) = r r d is the inverse of ( ) = d d r by showing that ( ( )) = r d r r and ( ( )) = d r d d. (c) Predict the speed that a car was traveling if the distance required to stop was 300 feet. 98. Height and Head Circumference The head circumference C of a child is related to the height H of the child (both in inches) through the function ( ) = − H C C 2.15 10.53 (a) Express the head circumference C as a function of height H. (b) Verify that ( ) = C C H is the inverse of ( ) = H H C by showing that ( ( )) = H C H H and ( ( )) = C H C C. (c) Predict the head circumference of a child who is 26 inches tall. 99. Devine Formula The Devine Formula, used to compute healthy body weight W for males (in kilograms) as a function of height h (in inches), is given by the function ( ) ( ) = + − W h h 50 2.3 60 (a) What is the healthy weight of a 6-foot male? (b) Express the height h as a function of weight W. (c) Verify that ( ) = h h W is the inverse of ( ) = W W h by showing that ( ( )) = h W h h and ( ( )) = W h W W. (d) What is the height of a male whose healthy weight of 80 kilograms? 100. Temperature Conversion The function ( ) = + F C C 9 5 32 converts a temperature from C degrees Celsius to F degrees Fahrenheit. (a) Express the temperature in degrees Celsius C as a function of the temperature in degrees Fahrenheit F. (b) Verify that ( ) = C C F is the inverse of ( ) = F f C by showing that ( ( )) = C F C C and ( ( )) = F C F F. (c) What is the temperature in degrees Celsius if it is 70 degrees Fahrenheit? 101. Income Taxes The function ( ) = + − T g g ( ) 5426 0.22 47,150 represents the 2024 federal income tax T (in dollars) due for a “single” filer whose modified adjusted gross income is g dollars, where < ≤ g 47,150 100,525. 108. Can a one-to-one function and its inverse be equal? What must be true about the graph of f for this to happen? Give some examples to support your conclusion. Explaining Concepts 109. Draw the graph of a one-to-one function that contains the points ( ) − − 2, 3, ( ) 0, 0 , and ( ) 1, 5 . Now draw the graph of its inverse. Compare your graph to those of other students. Discuss any similarities. What differences do you see?
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