Chapter 3, Problem 26T

Solution

The inverse of the function $y=\frac{1}{3}x+4$ .

  $f^{-1}\left(x\right)=3x-12$

Given information:

The function $y=\frac{1}{3}x+4$ .

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 $y=\frac{1}{3}x+4$ , first switch x and y and then solve further for y to get the required inverse function as shown below

  $\begin{array}{c}y=\frac{1}{3}x+4\\ x=\frac{1}{3}y+4\left(\text{Switch }x\text{and }y\right)\\ x-4=\frac{1}{3}y\\ 3\left(x-4\right)=y\\ 3x-12=y\\ \Rightarrow f^{-1}\left(x\right)=3x-12\end{array}$

Thus, the inverse of the given function is $f^{-1}\left(x\right)=3x-12$ .