Calculus with Concepts in Calculus

Denny Gulick, Robert Ellis

Chapter 7

Inverse Functions, L'Hopital’s Rule And Differential Equations - all with Video Answers

Educators

Section 1

Inverse Functions

Problem 1

Determine whether the given function has an inverse. If an inverse exists, give the domain and range of the inverse and graph the function and its inverse.
$$
f(x)=x^{5}
$$

Tyler Moulton

Tyler Moulton

Numerade Educator

Problem 2

Determine whether the given function has an inverse. If an inverse exists, give the domain and range of the inverse and graph the function and its inverse.
$$
f(x)=5 x^{7}+4 x^{3}
$$

Tyler Moulton

Numerade Educator

Problem 3

Determine whether the given function has an inverse. If an inverse exists, give the domain and range of the inverse and graph the function and its inverse.
$$
f(x)=-x^{8}
$$

Tyler Moulton

Numerade Educator

Problem 4

Determine whether the given function has an inverse. If an inverse exists, give the domain and range of the inverse and graph the function and its inverse.
$$
f(x)=4 \sqrt[5]{x}
$$

Tyler Moulton

Numerade Educator

Problem 5

Determine whether the given function has an inverse. If an inverse exists, give the domain and range of the inverse and graph the function and its inverse.
$$
f(t)=\sqrt{4-t}
$$

Tyler Moulton

Numerade Educator

Problem 6

Determine whether the given function has an inverse. If an inverse exists, give the domain and range of the inverse and graph the function and its inverse.
$$
f(t)=\sqrt{1-t^{2}}
$$

Tyler Moulton

Numerade Educator

Problem 7

Determine whether the given function has an inverse. If an inverse exists, give the domain and range of the inverse and graph the function and its inverse.
$$
f(x)=x+|x|
$$

Tyler Moulton

Numerade Educator

Problem 8

Determine whether the given function has an inverse. If an inverse exists, give the domain and range of the inverse and graph the function and its inverse.
$$
f(x)=x+\sin x
$$

Tyler Moulton

Numerade Educator

Problem 9

Determine whether the given function has an inverse. If an inverse exists, give the domain and range of the inverse and graph the function and its inverse.
$$
f(x)=x-\sin x
$$

Tyler Moulton

Numerade Educator

Problem 10

Determine whether the given function has an inverse. If an inverse exists, give the domain and range of the inverse and graph the function and its inverse.
$$
f(x)=x^{2}+\sin x
$$

Tyler Moulton

Numerade Educator

Problem 11

Determine whether the given function has an inverse. If an inverse exists, give the domain and range of the inverse and graph the function and its inverse.
$$
f(z)=\tan z
$$

Tyler Moulton

Numerade Educator

Problem 12

Determine whether the given function has an inverse. If an inverse exists, give the domain and range of the inverse and graph the function and its inverse.
$$
f(z)=\tan z \text { for }-\pi / 2<z<\pi / 2
$$

Tyler Moulton

Numerade Educator

Problem 13

Determine whether the given function has an inverse. If an inverse exists, give the domain and range of the inverse and graph the function and its inverse.
$$
g(t)=\ln (3-t)
$$

Tyler Moulton

Numerade Educator

Problem 14

Determine whether the given function has an inverse. If an inverse exists, give the domain and range of the inverse and graph the function and its inverse.
$$
g(t)=\ln t^{2}
$$

Tyler Moulton

Numerade Educator

Problem 15

Plot the graph of $f .$ From the graph determine whether $f$ appears to have an inverse.
$$
f(x)=x^{4}+2 x^{3}+2 x^{2}+x
$$

Carson Merrill

Numerade Educator

Problem 16

Plot the graph of $f .$ From the graph determine whether $f$ appears to have an inverse.
$$
f(x)=2 x^{3}+2 x^{2}+x
$$

Carson Merrill

Numerade Educator

Problem 17

Plot the graph of $f .$ From the graph determine whether $f$ appears to have an inverse.
$$
f(x)=\frac{x^{3}}{1+x^{2}}
$$

Carson Merrill

Numerade Educator

Problem 18

Plot the graph of $f .$ From the graph determine whether $f$ appears to have an inverse.
$$
f(x)=\frac{x^{3}-x}{1+x^{2}}
$$

Carson Merrill

Numerade Educator

Problem 19

Determine which of the functions $h, j$, and $k$ is the inverse of $f$, and which is the inverse of $g$.

Bobby Barnes

University of North Texas

Problem 20

Let $f$ be the function whose graph is shown in Figure $7.6$. Make a rough sketch of the graphs of $1 / f$ and $f^{-1}$ on the same coordinate system.

Harsh Gadhiya

Numerade Educator

Problem 21

Let $f$ be a periodic function; that is, suppose there is a positive number $a$ such that
$$
f(a+x)=f(x)
$$
for all $x$ in the domain of $f .$ Show that $f$ does not have an inverse.

James Kiss

Numerade Educator

Problem 22

Use Exercise 21 to show that the given function does not have an inverse.
$$
f(x)=\sin x-\cos x
$$

Heather Zimmers

Heather Zimmers

Numerade Educator

Problem 23

Use Exercise 21 to show that the given function does not have an inverse.
$$
f(x)=2 \tan 3 x-3 \cos 4 x+17
$$

James Kiss

Numerade Educator

Problem 24

Use Exercise 21 to show that the given function does not have an inverse.
$$
f(x)=x-[x]
$$

Erika Bustos

Numerade Educator

Problem 25

Find a formula for the inverse of the function.
$$
f(x)=-4 x^{3}-1
$$

Carlos Pinilla

Numerade Educator

Problem 26

Find a formula for the inverse of the function.
$$
f(x)=-2 x^{5}+\frac{9}{4}
$$

Bobby Barnes

University of North Texas

Problem 27

Find a formula for the inverse of the function.
$$
g(x)=\sqrt{1+x}
$$

Linda Hand

Numerade Educator

Problem 28

Find a formula for the inverse of the function.
$$
g(t)=\sqrt{3-2 t}
$$

Nick Johnson

Numerade Educator

Problem 29

Find a formula for the inverse of the function.
$$
k(t)=\frac{t-1}{t+1}
$$

Khushbu Rani

Numerade Educator

Problem 30

Find a formula for the inverse of the function.
$$
k(t)=\frac{3 t+5}{t-4}
$$

Nick Johnson

Numerade Educator

Problem 31

Find an interval on which $f$ has an inverse. (Hint: Find an interval on which $f^{\prime}>0$ or on which $f^{\prime}<0 .$ )
$$
f(x)=x^{2}-4
$$

Willis James

Numerade Educator

Problem 32

Find an interval on which $f$ has an inverse. (Hint: Find an interval on which $f^{\prime}>0$ or on which $f^{\prime}<0 .$ )
$$
f(x)=x^{2}-3 x+2
$$

Willis James

Numerade Educator

Problem 33

Find an interval on which $f$ has an inverse. (Hint: Find an interval on which $f^{\prime}>0$ or on which $f^{\prime}<0 .$ )
$$
f(x)=x^{3}-5 x+1
$$

Willis James

Numerade Educator

Problem 34

Find an interval on which $f$ has an inverse. (Hint: Find an interval on which $f^{\prime}>0$ or on which $f^{\prime}<0 .$ )
$$
f(x)=x^{3}+5 x+1
$$

Willis James

Numerade Educator

Problem 35

Find an interval on which $f$ has an inverse. (Hint: Find an interval on which $f^{\prime}>0$ or on which $f^{\prime}<0 .$ )
$$
f(x)=\frac{1}{1+x^{2}}
$$

Willis James

Numerade Educator

Problem 36

Find an interval on which $f$ has an inverse. (Hint: Find an interval on which $f^{\prime}>0$ or on which $f^{\prime}<0 .$ )
$$
f(x)=\frac{x}{1+x^{2}}
$$

Willis James

Numerade Educator

Problem 37

Find an interval on which $f$ has an inverse. (Hint: Find an interval on which $f^{\prime}>0$ or on which $f^{\prime}<0 .$ )
$$
f(x)=\cos x
$$

Willis James

Numerade Educator

Problem 38

Find an interval on which $f$ has an inverse. (Hint: Find an interval on which $f^{\prime}>0$ or on which $f^{\prime}<0 .$ )
$$
f(x)=\tan x
$$

Willis James

Numerade Educator

Problem 39

Find an interval on which $f$ has an inverse. (Hint: Find an interval on which $f^{\prime}>0$ or on which $f^{\prime}<0 .$ )
$$
f(x)=\sin ^{2} x
$$

Willis James

Numerade Educator

Problem 40

Find an interval on which $f$ has an inverse. (Hint: Find an interval on which $f^{\prime}>0$ or on which $f^{\prime}<0 .$ )
$$
f(x)=\sec x
$$

Willis James

Numerade Educator

Problem 41

Use (8) to calculate $\left(f^{-1}\right)^{\prime}(c) .$ (Hint: Find $a$ by inspection.)
$$
f(x)=x^{3}+7 ; c=6
$$

Darshan Maheshwari

Darshan Maheshwari

Numerade Educator

Problem 42

Use (8) to calculate $\left(f^{-1}\right)^{\prime}(c) .$ (Hint: Find $a$ by inspection.)
$$
f(x)=5 x^{5}+4 x^{3} ; c=9
$$

Fuzail Shakir

Numerade Educator

Problem 43

Use (8) to calculate $\left(f^{-1}\right)^{\prime}(c) .$ (Hint: Find $a$ by inspection.)
$$
f(x)=x+\sin x ; c=0
$$

Linda Hand

Numerade Educator

Problem 44

Use (8) to calculate $\left(f^{-1}\right)^{\prime}(c) .$ (Hint: Find $a$ by inspection.)
$$
f(x)=x+\sqrt{x} ; c=2
$$

Adrian Co

Numerade Educator

Problem 45

Use (8) to calculate $\left(f^{-1}\right)^{\prime}(c) .$ (Hint: Find $a$ by inspection.)
$$
f(x)=4 \ln x ; c=0
$$

Gregory Higby

Numerade Educator

Problem 46

Use (8) to calculate $\left(f^{-1}\right)^{\prime}(c) .$ (Hint: Find $a$ by inspection.)
$$
f(x)=\tan x \text { for }-\pi / 2<x<\pi / 2 ; c=\sqrt{3}
$$

Lucas Finney

Numerade Educator

Problem 47

Use (8) to calculate $\left(f^{-1}\right)^{\prime}(c) .$ (Hint: Find $a$ by inspection.)
$$
f(t)=3 t-\left(1 / t^{3}\right) \text { for } t<0 ; c=-2
$$

Darshan Maheshwari

Darshan Maheshwari

Numerade Educator

Problem 48

Use (8) to calculate $\left(f^{-1}\right)^{\prime}(c) .$ (Hint: Find $a$ by inspection.)
$$
f(t)=t \ln t ; c=2 e^{2}
$$

Darshan Maheshwari

Darshan Maheshwari

Numerade Educator

Problem 49

Find $d x / d y$.
$$
y=x^{9}+7 x
$$

Paul Teng

Numerade Educator

Problem 50

Find $d x / d y$.
$$
y=x-2 / x \text { for } x<0
$$

MM

Matthys Marthinus

Numerade Educator

Problem 51

Find $d x / d y$.
$$
y=\ln \left(x^{3}+1\right)
$$

Amy Jiang

Numerade Educator

Problem 52

Find $d x / d y$.
$$
y=x+\cos x
$$

Joshua Eastwood

Joshua Eastwood

Numerade Educator

Problem 53

Find $d x / d y$.
$$
y=\sin x \text { for }-\pi / 2<x<\pi / 2
$$

Taylor Shimono

Numerade Educator

Problem 54

Find $d x / d y$.
$$
y=\tan x \text { for }-\pi / 2<x<\pi / 2
$$

Eric Sigman

Numerade Educator

Problem 55

Suppose the graph of the derivative of a function $f$ is as shown in Figure 7.7. On which subintervals of $[1,9]$ does $f$ have an inverse?

Amrita Bhasin

Numerade Educator

Problem 56

Suppose $f$ has a continuous derivative on the interval $[0,6]$. Assume also that $f^{\prime}$ is increasing on $[0,4], f^{\prime}$ is decreasing
on $[4,6]$, and
$f^{\prime}(0)=-1, \quad f^{\prime}(3)=0, \quad f^{\prime}(4)=2, \quad$ and $\quad f^{\prime}(5)=0$
On which subintervals of $[0,6]$ does $f$ have an inverse?

Saad Ali Khan

Numerade Educator

Problem 57

Let
$$
f(x)=\int_{0}^{x} \sqrt{1+t^{4}} d t \quad \text { for all } x
$$
a. Show that $f$ has an inverse.
b. Let $c=f(1)$. Find $\left(f^{-1}\right)^{\prime}(c) .$ (Hint: Do not attempt to evaluate $f(1) .)$

Gregory Higby

Numerade Educator

Problem 58

Let
$$
f(x)=\int_{0}^{x^{3}} \sin ^{6}\left(t^{2}\right) d t \quad \text { for all } x
$$
a. Show that $f$ has an inverse.
b. Let $c=f(\sqrt[6]{\pi / 6})$. Find $\left(f^{-1}\right)^{\prime}(c)$.

Yaw Asomani

Numerade Educator

Problem 59

a. Can a polynomial of even degree have an inverse? Explain.
b. Can a polynomial of odd degree have an inverse? Explain.

Vishal Parmar

Numerade Educator

Problem 60

Assume that $f$ has an inverse.
a. Suppose the graph of $f$ lies in the first quadrant. In which quadrant does the graph of $f^{-1}$ lie?
b. Suppose the graph of $f$ lies in the second quadrant. In which quadrant does the graph of $f^{-1}$ lie?

Yujie Wang

College of San Mateo

Problem 61

Show that if $f$ and $g$ both have inverses, then $g \circ f$ has an
inverse and
$$
(g \circ f)^{-1}=f^{-1} \circ g^{-1}
$$

Yaw Asomani

Numerade Educator

Problem 62

Using (7), along with Exercises 1, 29 , and 61 , show that the following functions have inverses.
a. $k(x)=\left(\frac{x-1}{x+1}\right)^{5}$
b. $k(x)=\ln \left(\frac{x-1}{x+1}\right)$
c. $k(x)=\frac{\ln x-1}{\ln x+1}$

Gaurav Kalra

Numerade Educator

Problem 63

Assume that $g$ has an inverse, and let $f(x)=-x$. Using Exercise 61 , show that
$$
(g \circ f)^{-1}(x)=-g^{-1}(x)
$$
for all $x$ in the domain of $g^{-1}$.

Savannah Payne

Numerade Educator

Problem 64

a. Show that the inverse of an increasing function is increasing.
b. Show that the inverse of a decreasing function is decreasing.

Gaurav Kalra

Numerade Educator

Problem 65

Assume that $f$ has an inverse, and let $a$ be a fixed number.
Let
$$
g(x)=f(x+a)
$$
for all $x$ such that $x+a$ is in the domain of $f$. Show that $g$ has an inverse and that $g^{-1}(x)=f^{-1}(x)-a$.

James Kiss

Numerade Educator

Problem 66

Assume that $f$ has an inverse, and let $a$ be a fixed number different from 0 . Let
$$
g(x)=f(a x)
$$
for all $x$ such that $a x$ is in the domain of $f$. Show that $g$ has an inverse and that $g^{-1}(x)=f^{-1}(x) / a$.

James Kiss

Numerade Educator

Problem 67

Let $y=x^{3}$. Then
$$
\frac{d y}{d x}=3 x^{2} \quad \text { and } \quad \frac{d x}{d y}=\frac{1}{3 y^{2 / 3}}
$$
Show that equation (10) holds for this function if we evaluate $d y / d x$ at $x=2$ and $d x / d y$ at $y=8$ but that it does

Matt Just

Numerade Educator

Problem 68

Let $f$ be a function with an inverse, and suppose $f^{\prime}(a)=0$. Show that $f^{-1}$ is not differentiable at $f(a)$. (Hint: Prove by contradiction, using the Chain Rule and differentiating the equation $f^{-1}(f(x))=x$ implicitly. $)$

Linda Hand

Numerade Educator

Problem 69

Using Exercise 68 , show that the following functions are not differentiable at the given value of $c$.
a. $f^{-1}$, where $f(x)=x+\sin x ; c=\pi$.
b. $f^{-1}$, where $f(x)=x^{5}+x^{3}-4 ; c=-4$.

SN

Shumayal N

Numerade Educator

Problem 70

Let $0 \leq a<b$, and let $f$ be nonnegative, increasing, and continuous on $[a, b]$, so that $f^{-1}$ exists. Let $A_{1}$ and $A_{2}$ be the areas of the regions $R_{1}$ and $R_{2}$, respectively, in Figure $7.8$.
a. Show that $A_{1}+A_{2}=b f(b)-a f(a)$.
b. Use (a) to prove that $\int_{a}^{b} f(x) d x=b f(b)-a f(a)-\int_{f(a)}^{f(b)} f^{-1}(y) d y$

Uma Kumari

Numerade Educator

Problem 71

Use Exercise $70(\mathrm{~b})$ to evaluate $\int_{0}^{1}\left[(x-1)^{1 / 3}+1\right]^{1 / 2} d x$.

Gregory Higby

Numerade Educator

Problem 72

a. Let $f(x)=-\sqrt{x}$ for $x>0$. Show that the graphs of $f$ and $f^{-1}$ are concave upward on their respective domains.
b. Let $f(x)=\sqrt{x}$ for $x>0$. Show that the graph of $f$ is concave downward on $(0, \infty)$, whereas the graph of $f^{-1}$ is concave upward on $(0, \infty)$.

Carson Merrill

Numerade Educator

Problem 73

Assume that $f$ has an inverse, that $f^{\prime \prime}\left(f^{-1}(x)\right)$ exists, and that $f^{\prime}\left(f^{-1}(x)\right) \neq 0$. Show that
$$
(f-1)^{\prime \prime}(x)=\frac{-f^{\prime \prime}\left(f^{-1}(x)\right)}{\left[f^{\prime}\left(f^{-1}(x)\right)\right]^{3}}
$$

Amy Jiang

Numerade Educator

Problem 74

Let the domain of $f$ be an open interval $I$, and assume that $f^{-1}$ exists. Suppose $f^{\prime \prime}$ exists on $I$ and $f^{\prime}(x) \neq 0$ and $f^{\prime \prime}(x) \neq 0$ for all $x$ in $I$
a. Suppose $f$ is increasing on $I$. Show that the graph of $f^{-1}$ is concave upward on its domain if the graph of $f$ is concave downward on $I$, and the graph of $f^{-1}$ is concave downward on its domain if the graph of $f$ is concave upward on $I .$ (Hint: Use Exercise 73.)
b. Suppose $f$ is decreasing on $I$. Show that the graph of $f^{-1}$ is concave upward on its domain if the graph of $f$ is concave upward on $I$, and the graph of $f^{-1}$ is concave downward on its domain if the graph of $f$ is concave downward on $I$.
c. Suppose that $a$ is in $I$ and that the graph of $f$ has an inflection point at $(a, f(a))$. What can you say about an inflection point for the graph of $f^{-1}$ ?

Carson Merrill

Numerade Educator

Problem 75

Let $f(x)=x^{2 / 3}$ for $0 \leq x \leq 8$. Find the length $L$ of the graph of $f$. (Hint: Since $f^{\prime}(0)$ does not exist, the formula in Section $6.2$ for length does not apply. However, $L$ equals the length of the graph of $f^{-1}$.)

Tyler Moulton

Numerade Educator

Problem 76

Suppose a person is traversing a path that has the shape of the graph of $y=x^{2 / 3}$ from the point $(0,0)$ to the point $(1,1)$. Find the halfway point on the curve. (Hint: Use the idea of the solution of Exercise $75 .$ )

Monica Miller

Numerade Educator

Problem 77

Let $f$ be the function representing the conversion from inches to centimeters, and let $g$ be the function representing the conversion from centimeters to inches.
Then
$$
f(x)=2.54 x \text { for } x \geq 0
$$
and
$$
g(x)=\frac{1}{2.54} x \text { for } x \geq 0
$$
Show that $f$ and $g$ are inverses of one another.

Shannon O'Connor

Shannon O'Connor

Numerade Educator

Problem 78

Let
$$
f(v)=\frac{m_{0}}{\sqrt{1-v^{2} / c^{2}}} \text { for } v \geq 0
$$
and
$$
g(m)=c \sqrt{1-m_{0}^{2} / m^{2}} \quad \text { for } m \geq m_{0}
$$
where $m_{0}$ and $c$ are constants. Show that $g$ is the inverse of $f$. (The functions $f$ and $g$ arise in the theory of relativity. If $c$ is the speed of light in a vacuum and $m_{0}$ is the rest mass of a particle, then $f(v)$ is the mass of the particle as it moves with velocity $v$, and $g(m)$ is the velocity of the particle when it has mass $m$.)

Jamie M

Numerade Educator