Chapter EP, Problem 7.2.12EP

To determine

To calculate: The inverse of the functions $y=\frac{1}{2}x-1$ . Also graph the function and its inverse.

Answer to Problem 7.2.12EP

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

The red line depicts the function $y=\frac{1}{2}x-1$ and blue line depicts its inverse $f^{-1}\left(x\right)=2x+2$ .

Explanation of Solution

Given information:

The functions $y=\frac{1}{2}x-1$ .

Formula used:

If two functions f and $f^{-1}$ are said to be inverse functions then $f\left(a\right)=b$ if and only if $f^{-1}\left(b\right)=a$ .

Calculation:

Consider the functions $y=\frac{1}{2}x-1$ .

Let $f\left(x\right)=y$ .

Interchange the position of x and y ,

  $x=\frac{1}{2}y-1$

Evaluate the above equation for y ,

  $\begin{array}{c}x=\frac{1}{2}y-1\\ \frac{1}{2}y=x+1\\ y=2x+2\end{array}$

Replace yby $f^{-1}\left(x\right)$ ,

  $f^{-1}\left(x\right)=2x+2$

Graph of the function $y=\frac{1}{2}x-1$ and its inverse $f^{-1}\left(x\right)=2x+2$ is provided below,

The red line depicts the function $y=\frac{1}{2}x-1$ and blue line depicts its inverse $f^{-1}\left(x\right)=2x+2$ .

It is observed that the graph of $f^{-1}\left(x\right)=2x+2$ is the reflection of the graph $y=\frac{1}{2}x-1$ in the line $y=x$ (the green dotted line in the graph).

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