Chapter EP, Problem 7.2.11EP

To determine

To calculate: The inverse of the functions $y=\frac{x-5}{3}$ . Also graph the function and its inverse.

Answer to Problem 7.2.11EP

The inverse of the function $y=\frac{x-5}{3}$ is $f^{-1}\left(x\right)=3x+5$ .

The red line depicts the function $y=\frac{x-5}{3}$ and blue line depicts its inverse $f^{-1}\left(x\right)=3x+5$ .

Explanation of Solution

Given information:

The functions $y=2x+8$ .

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 $y=2x+8$ .

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

Interchange the position of x and y ,

  $x=\frac{y-5}{3}$

Evaluate the above equation for y ,

  $\begin{array}{c}x=\frac{y-5}{3}\\ 3x=y-5\\ y=3x+5\end{array}$

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

  $f^{-1}\left(x\right)=3x+5$

Graph of the function $y=\frac{x-5}{3}$ and its inverse $f^{-1}\left(x\right)=3x+5$ is provided below,

The red line depicts the function $y=\frac{x-5}{3}$ and blue line depicts its inverse $f^{-1}\left(x\right)=3x+5$ .

It is observed that the graph of $f^{-1}\left(x\right)=3x+5$ is the reflection of the graph $y=\frac{x-5}{3}$ in the line $y=x$ (the green dotted line in the graph).

Thus, the inverse of the function $y=\frac{x-5}{3}$ is $f^{-1}\left(x\right)=3x+5$ .