Chapter 3.4, Problem 25E

To determine

To find: $f^{-1}\left(x\right)$ and state whether it is a function.

Answer to Problem 25E

  $f^{-1}\left(x\right)=\frac{x-7}{2}$

Yes, $f^{-1}\left(x\right)$ is a function.

Explanation of Solution

Given information:

  $f\left(x\right)=2x+7$

Calculation:

The $f^{-1}\left(x\right)$ can be evaluated as follows:

  $\begin{array}{c}y=2x+7\left[\text{Let }f\left(x\right)=y\right]\\ x=2y+7\left[\text{Interchange }x\text{ and }y\right]\\ 2y=x-7\left[\text{Solve for }y\right]\\ y=\frac{x-7}{2}\\ f^{-1}\left(x\right)=\frac{x-7}{2}\left[\text{Replace }y\text{ with }{f}^{-1}\left(x\right)\right]\end{array}$

Since there is only one value of y for every value of x.Thus, the $f^{-1}\left(x\right)=\frac{x-7}{2}$ is a function