Chapter 7.2, Problem 17PPS

To determine

The inverse of the function $y=-2x+1$ and graph the function and its inverse.

Answer to Problem 17PPS

The inverse of function $y=-2x+1$ where $y=f\left(x\right)$ is $f^{-1}\left(x\right)=\frac{1-x}{2}$ .

Explanation of Solution

Given information: The given function is,

  $y=-2x+\mathrm{1......}\left(\text{where }y=f\left(x\right)\right)$

Concept used: In order to find the inverse of a given function $f\left(x\right)$ write $f\left(x\right)=y$ and exchange the variables $\left(x\right)\text{ and }\left(y\right)$ and solve for the variable $\left(y\right)$ . At last write $y=f^{-1}\left(x\right)$ which is the required inverse of the function $f\left(x\right)$ .

Calculation: For the given function $y=-2x+1$ , the inverse is evaluated as,

  $\begin{array}{c}y=-2x+1\\ \text{or, }\left(x\right)=-2\left(y\right)+\mathrm{1......}\left(\text{Exchanging variables}\right)\\ \text{or, }y=\frac{1-x}{2}\mathrm{......}\left(\text{Solving for variable }\left(y\right)\right)\\ f^{-1}\left(x\right)=\frac{1-x}{2}\mathrm{......}\left(\text{Plugging }y=f^{-1}\left(x\right)\right)\end{array}$

Hence, the inverse of function $y=-2x+1$ is $f^{-1}\left(x\right)=\frac{1-x}{2}$ .

Graph: The graph for the function $y=-2x+1$ and its inverse function $f^{-1}\left(x\right)=\frac{1-x}{2}$ is shown as,