Chapter 3.4, Problem 9CFU

To determine

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

Answer to Problem 9CFU

  $f^{-1}\left(x\right)=\frac{-x+2}{3}$

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

Explanation of Solution

Given information:

  $f\left(x\right)=-3x+2$

Calculation:

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

  $\begin{array}{c}y=-3x+2\left[\text{let }f\left(x\right)=y\right]\\ x=-3y+2\left[\text{Interchange }x\text{ and }y\right]\\ x+3y=2\left[\text{Solve for }y\right]\\ 3y=-x+2\\ y=\frac{-x+2}{3}\\ f^{-1}\left(x\right)=\frac{-x+2}{3}\left[\text{Replace }y\text{ with }{f}^{-1}\left(x\right)\right]\end{array}$

Since there is only one value of y for every value of x.Thus, the $f^{-1}\left(x\right)=\frac{-x+2}{3}$ is a function.