Chapter 3.4, Problem 45E

Solution

To show that the inverse of a linear function $f\left(x\right)=mx+b$ is also a linear function.

The slope of the inverse function is $\frac{1}{m}$ and the y -intercept is $-\frac{b}{m}$

Given information:

The linear function $f\left(x\right)=mx+b$ , where $m\ne 0$ .

Concept used:

• The graph of the function $y=f\left(x\right)$ and its inverse $f^{-1}\left(x\right)$ are symmetric to the line $y=x$ .
• For any point $\left(a,b\right)$ on the graph of $f\left(x\right)$ , the corresponding point on the graph of inverse function $f^{-1}\left(x\right)$ is $\left(b,a\right)$ .
• If a function is one-to-one, then it has an inverse.

Calculation:

In order to find the inverse of the function $f\left(x\right)=mx+b$ , let $y=f\left(x\right)$ , so the given function becomes

  $y=mx+b$

Now to find the inverse function, first interchange the x and y and then solve for y as shown below

  $\begin{array}{c}y=mx+b\\ x=my+b\left(\text{interchang }x\text{ and }y\right)\\ x-b=my\left(\text{Solve for }y\right)\\ \frac{1}{m}\cdot x-\frac{1}{m}\cdot b=y\\ \frac{1}{m}\cdot x-\frac{b}{m}=y\\ \Rightarrow f^{-1}\left(x\right)=\frac{1}{m}x-\frac{b}{m}\end{array}$

It implies that the inverse of the given function is also a linear function.

On comparing with the slope-intercept form $y=mx+b$ , the slope of the inverse function is $\frac{1}{m}$ and the y -intercept is $-\frac{b}{m}$ .