Calculus

James Stewart

Chapter 6

Inverse Functions: - all with Video Answers

Educators

Section 1

6.1 Exercises

Problem 1

(a) What is a one-to-one function?
(b) How can you tell from the graph of a function whether it is one-to-one?

Amy Jiang

Amy Jiang

Numerade Educator

Problem 2

(a) Suppose $f$ is a one-to-one function with domain $A$ and range $B$. How is the inverse function $f^{-1}$ defined? What is the domain of $f^{-1}$ ? What is the range of $f^{-1}$ ?
(b) If you are given a formula for $f$, how do you find a formula for $f^{-1}$ ?
(c) If you are given the graph of $f$, how do you find the graph of $f^{-1}$ ?

Jeffrey Payo

Numerade Educator

Problem 3

A function is given by a table of values, a graph, a formula, or a verbal description. Determine whether it is one-to-one.
(Table Can't Copy)

Carson Merrill

Numerade Educator

Problem 4

A function is given by a table of values, a graph, a formula, or a verbal description. Determine whether it is one-to-one.
(Table Can't Copy)

Carson Merrill

Numerade Educator

Problem 5

A function is given by a table of values, a graph, a formula, or a verbal description. Determine whether it is one-to-one.
(Graph Can't Copy)

Carson Merrill

Numerade Educator

Problem 6

A function is given by a table of values, a graph, a formula, or a verbal description. Determine whether it is one-to-one.
(Graph Can't Copy)

Carson Merrill

Numerade Educator

Problem 7

A function is given by a table of values, a graph, a formula, or a verbal description. Determine whether it is one-to-one.
(Graph Can't Copy)

Carson Merrill

Numerade Educator

Problem 8

A function is given by a table of values, a graph, a formula, or a verbal description. Determine whether it is one-to-one.
(Graph Can't Copy)

Carson Merrill

Numerade Educator

Problem 9

A function is given by a table of values, a graph, a formula, or a verbal description. Determine whether it is one-to-one.
$f(x)=x^2-2 x$

Amrita Bhasin

Numerade Educator

Problem 11

A function is given by a table of values, a graph, a formula, or a verbal description. Determine whether it is one-to-one.
$g(x)=1 / x$

Amrita Bhasin

Numerade Educator

Problem 13

A function is given by a table of values, a graph, a formula, or a verbal description. Determine whether it is one-to-one.
$h(x)=1+\cos x$

William Macmillan

William Macmillan

Numerade Educator

Problem 14

A function is given by a table of values, a graph, a formula, or a verbal description. Determine whether it is one-to-one.
$h(x)=1+\cos x, \quad 0 \leqslant x \leqslant \pi$

William Macmillan

William Macmillan

Numerade Educator

Problem 15

A function is given by a table of values, a graph, a formula, or a verbal description. Determine whether it is one-to-one.
$f(t)$ is the height of a football $t$ seconds after kickoff.

Amrita Bhasin

Numerade Educator

Problem 16

A function is given by a table of values, a graph, a formula, or a verbal description. Determine whether it is one-to-one.
$f(t)$ is your height at age $t$.

Carlos Pinilla

Numerade Educator

Problem 17

Assume that $f$ is a one-to-one function.
(a) If $f(6)=17$, what is $f^{-1}(17)$ ? (b) If $f^{-1}(3)=2$, what is $f(2)$ ?

Victoria Dollar

Victoria Dollar

Numerade Educator

Problem 18

If $f(x)=x^5+x^3+x$, find $f^{-1}(3)$ and $f\left(f^{-1}(2)\right)$.

Jeffrey Payo

Numerade Educator

Problem 19

If $h(x)=x+\sqrt{x}$, find $h^{-1}(6)$.

Brian Austin

Numerade Educator

Problem 20

The graph of $f$ is given.
(a) Why is $f$ one-to-one?
(b) What are the domain and range of $f^{-1}$ ?
(c) What is the value of $f^{-1}(2)$ ?
(d) Estimate the value of $f^{-1}(0)$.

Carson Merrill

Numerade Educator

Problem 21

The formula $C=\frac{5}{5}(F-32)$, where $F \supsetneqq-459.67$, expresses the Celsius temperature $C$ as a function of the Fahrenheit temperature $F$. Find a formula for the inverse function and interpret it. What is the domain of the inverse function?

Nicholas Carlson

Nicholas Carlson

Numerade Educator

Problem 22

In the theory of relativity, the mass of a particle with speed $v$ is $$ m=f(v)=\frac{m_0}{\sqrt{1-v^2 / c^2}} $$ where $m_0$ is the rest mass of the particle and $c$ is the speed of light in a vacuum. Find the inverse function of $f$ and explain its meaning.

Carlos Pinilla

Numerade Educator

Problem 23

Find a formula for the inverse of the function.
$f(x)=3-2 x$

Sheryl Ezze

Numerade Educator

Problem 24

Find a formula for the inverse of the function.
$f(x)=\frac{4 x-1}{2 x+3}$

Heather Zimmers

Heather Zimmers

Numerade Educator

Problem 25

Find a formula for the inverse of the function.
$f(x)=1+\sqrt{2+3 x}$

Amrita Bhasin

Numerade Educator

Problem 26

Find a formula for the inverse of the function.
$y=x^2-x, \quad x \geqslant \frac{1}{2}$

Heather Zimmers

Heather Zimmers

Numerade Educator

Problem 27

Find a formula for the inverse of the function.
$y=\frac{1-\sqrt{x}}{1+\sqrt{x}}$

Amrita Bhasin

Numerade Educator

Problem 28

Find a formula for the inverse of the function.
$f(x)=2 x^2-8 x, \quad x \geqslant 2$

Linda Hand

Numerade Educator

Problem 29

Find an explicit formula for $f^{-1}$ and use it to graph $f^{-1}, f$, and the line $y=x$ on the same screen. To check your work, see whether the graphs of $f$ and $f^{-1}$ are reflections about the line.
$f(x)=x^4+1, \quad x \geqslant 0$

Amrita Bhasin

Numerade Educator

Problem 30

Find an explicit formula for $f^{-1}$ and use it to graph $f^{-1}, f$, and the line $y=x$ on the same screen. To check your work, see whether the graphs of $f$ and $f^{-1}$ are reflections about the line.
$f(x)=\sqrt{x^2+2 x}, \quad x>0$

Heather Zimmers

Heather Zimmers

Numerade Educator

Problem 31

Use the given graph of $f$ to sketch the graph of $f^{-1}$.
(Graph Can't Copy)

James Kiss

Numerade Educator

Problem 32

Use the given graph of $f$ to sketch the graph of $f^{-1}$.
(Graph Can't Copy)

Check back soon!

Problem 33

Let $f(x)=\sqrt{1-x^2}, 0 \leqslant x \leqslant 1$.
(a) Find $f^{-1}$. How is it related to $f$ ?
(b) Identify the graph of $f$ and explain your answer to part (a).

Victor Salazar

Numerade Educator

Problem 34

Let $g(x)=\sqrt[3]{1-x^3}$.
(a) Find $g^{-1}$. How is it related to $g$ ?
(b) Graph $g$. How do you explain your answer to part (a)?

James Kiss

Numerade Educator

Problem 35

(a) Show that $f$ is one-to-one.
(b) Use Theorem 7 to find $\left(f^{-1}\right)^{\prime}(a)$.
(c) Calculate $f^{-1}(x)$ and state the domain and range of $f^{-1}$.
(d) Calculate $\left(f^{-1}\right)^{\prime}(a)$ from the formula in part (c) and check that it agrees with the result of part (b).
(e) Sketch the graphs of $f$ and $f^{-1}$ on the same axes.
$f(x)=x^3, \quad a=8$

Amy Jiang

Numerade Educator

Problem 36

(a) Show that $f$ is one-to-one.
(b) Use Theorem 7 to find $\left(f^{-1}\right)^{\prime}(a)$.
(c) Calculate $f^{-1}(x)$ and state the domain and range of $f^{-1}$.
(d) Calculate $\left(f^{-1}\right)^{\prime}(a)$ from the formula in part (c) and check that it agrees with the result of part (b).
(e) Sketch the graphs of $f$ and $f^{-1}$ on the same axes.
$f(x)=\sqrt{x-2}, \quad a=2$

Amy Jiang

Numerade Educator

Problem 37

(a) Show that $f$ is one-to-one.
(b) Use Theorem 7 to find $\left(f^{-1}\right)^{\prime}(a)$.
(c) Calculate $f^{-1}(x)$ and state the domain and range of $f^{-1}$.
(d) Calculate $\left(f^{-1}\right)^{\prime}(a)$ from the formula in part (c) and check that it agrees with the result of part (b).
(e) Sketch the graphs of $f$ and $f^{-1}$ on the same axes.
$f(x)=9-x^2, \quad 0 \leqslant x \leqslant 3, \quad a=8$

Amy Jiang

Numerade Educator

Problem 38

(a) Show that $f$ is one-to-one.
(b) Use Theorem 7 to find $\left(f^{-1}\right)^{\prime}(a)$.
(c) Calculate $f^{-1}(x)$ and state the domain and range of $f^{-1}$.
(d) Calculate $\left(f^{-1}\right)^{\prime}(a)$ from the formula in part (c) and check that it agrees with the result of part (b).
(e) Sketch the graphs of $f$ and $f^{-1}$ on the same axes.
$f(x)=1 /(x-1), \quad x>1, \quad a=2$

Amy Jiang

Numerade Educator

Problem 39

Find $\left(f^{-1}\right)^{\prime}(a)$.
$f(x)=2 x^3+3 x^2+7 x+4, \quad a=4$

Linda Hand

Numerade Educator

Problem 40

Find $\left(f^{-1}\right)^{\prime}(a)$.
$f(x)=x^3+3 \sin x+2 \cos x, \quad a=2$

Linda Hand

Numerade Educator

Problem 41

Find $\left(f^{-1}\right)^{\prime}(a)$.
$f(x)=3+x^2+\tan (\pi x / 2), \quad-1<x<1, \quad a=3$

Hoan Nguyen

Numerade Educator

Problem 42

Find $\left(f^{-1}\right)^{\prime}(a)$.
$f(x)=\sqrt{x^3+x^2+x+1}, \quad a=2$

Linda Hand

Numerade Educator

Problem 43

Suppose $f^{-1}$ is the inverse function of a differentiable function $f$ and $f(4)=5, f^{\prime}(4)=\frac{2}{3}$. Find $\left(f^{-1}\right)^{\prime}(5)$.

Amy Jiang

Numerade Educator

Problem 44

If $g$ is an increasing function such that $g(2)=8$ and $g^{\prime}(2)=5$, calculate $\left(g^{-1}\right)^{\prime}(8)$.

Amy Jiang

Numerade Educator

Problem 45

If $f(x)=\int_3^x \sqrt{1+t^3} d t$, find $\left(f^{-1}\right)^{\prime}(0)$.

Vishal Parmar

Numerade Educator

Problem 46

Suppose $f^{-1}$ is the inverse function of a differentiable function $f$ and let $G(x)=1 / f^{-1}(x)$. If $f(3)=2$ and $f^{\prime}(3)=\frac{1}{9}$, find $G^{\prime}(2)$.

Umar Sohail Qureshi

Umar Sohail Qureshi

Numerade Educator

Problem 48

Show that $h(x)=\sin x, x \in \mathbb{R}$, is not one-to-one, but its restriction $f(x)=\sin x,-\pi / 2 \leqslant x \leqslant \pi / 2$, is one-to-one. Compute the derivative of $f^{-1}=\sin ^{-1}$ by the method of Note 2.

Gregory Higby

Numerade Educator

Problem 49

(a) If we shift a curve to the left, what happens to its reflection about the line $y=x$ ? In view of this geometric principle, find an expression for the inverse of $g(x)=f(x+c)$, where $f$ is a one-to-one function,
(b) Find an expression for the inverse of $h(x)=f(c x)$, where $c \neq 0$.

Victoria Dollar

Victoria Dollar

Numerade Educator

Problem 50

(a) If $f$ is a one-to-one, twice differentiable function with inverse function $g$, show that
$$ g^{\prime \prime}(x)=-\frac{f^{\prime \prime}(g(x))}{\left[f^{\prime}(g(x))\right]^3} $$
(b) Deduce that if $f$ is increasing and concave upward, then its inverse function is concave downward.

Linda Hand

Numerade Educator