Chapter 3.4, Problem 14E

Solution

The function f such that the graph of inverse function is a line with a slope of 3.

  $f\left(x\right)=\frac{1}{3}x+2$

Given information:

The inverse function $f^{-1}$ has a slope of 3.

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 function f such that its inverse $f^{-1}$ has a slope of 3, the function must have the coefficient of x as $\frac{1}{3}$ in the function f .

Once such function is $f\left(x\right)=\frac{1}{3}x+2$ . The inverse of this function is

  $\begin{array}{c}f\left(x\right)=\frac{1}{3}x+2\\ y=\frac{1}{3}x+2\\ x=\frac{1}{3}y+2\left(\text{interchang }x\text{ and }y\right)\\ x-2=\frac{1}{3}y\left(\text{Solve for }y\right)\\ 3x-6=y\\ \Rightarrow f^{-1}\left(x\right)=3x-6\end{array}$

Thus, on comparing with the slope-intercept form $y=mx+b$ , the slope of the inverse function is 3.