Chapter 1.6, Problem 2E

(a)

To determine

To define: The inverse function $f^{-1}$; what is the domain and range of $f^{-1}$.

Explanation of Solution

Given:

The function f is one-to-one with domain A and range B.

Calculation:

The inverse function is defined as $y=f^{-1}\left(x\right)$.

Let $f^{-1}$ be a one-to-one function with domain B and range A.

The domain of $f^{-1}$ is the range of f and the range of $f^{-1}$ is the domain of f.

Thus, the domain of $f^{-1}$ is B and the range of $f^{-1}$ is A.

(b)

To determine

To Find: The formula for $f^{-1}$ if the formula for f is given.

Answer to Problem 2E

The formula for $f^{-1}$ is $y=f^{-1}\left(x\right)$.

Explanation of Solution

Let $f\left(x\right)=y$ be the given function.

To get the formula of $f^{-1}$, multiply $f^{-1}$ on both sides.

$\begin{array}{c}{f}^{-1}f\left(x\right)=f^{-1}\left(y\right)\\ x=f^{-1}\left(y\right)\end{array}$

Interchange x and y,

$y=f^{-1}\left(x\right)$

The formula for $f^{-1}$ is $y=f^{-1}\left(x\right)$.

(c)

To determine

To explain: How to obtain the graph of $f^{-1}$ if the graph of f is given f.

Explanation of Solution

Given that the graph of $f\left(x\right)$.

First check whether the graph of the function f is one-to-one by the horizontal line test.

If the function f is not one-to-one then $f^{-1}$ does not exist.

If the function f is one-to-one then the graph of $f^{-1}$ is the reflection of the graph f about the line $y=x$