Chapter 7.2, Problem 18PPS

To determine

The inverse of the function $h\left(x\right)=\frac{x-4}{3}$ and graph the function and its inverse.

Answer to Problem 18PPS

The inverse of function $h\left(x\right)=\frac{x-4}{3}$ is $h^{-1}\left(x\right)=3x+4$ .

Explanation of Solution

Given information: The given function is,

  $h\left(x\right)=\frac{x-4}{3}$

Concept used: In order to find the inverse of a given function $f\left(x\right)$ write $f\left(x\right)=y$ and exchange the variables $\left(x\right)\text{ and }\left(y\right)$ and solve for the variable $\left(y\right)$ . At last write $y=f^{-1}\left(x\right)$ which is the required inverse of the function $f\left(x\right)$ .

Calculation: For the given function $h\left(x\right)=\frac{x-4}{3}$ , the inverse is evaluated as,

  $\begin{array}{c}y=\frac{x-4}{3}\mathrm{......}\left(\text{Plugging }h\left(x\right)=y\right)\\ \text{or, }\left(x\right)=\frac{\left(y\right)-4}{3}\mathrm{......}\left(\text{Exchanging variables}\right)\\ y=3x+\mathrm{4......}\left(\text{Solving for variable }\left(y\right)\right)\\ h^{-1}\left(x\right)=3x+\mathrm{4......}\left(\text{Plugging }y=h^{-1}\left(x\right)\right)\end{array}$

Hence, the inverse of function $h\left(x\right)=\frac{x-4}{3}$ is $h^{-1}\left(x\right)=3x+4$ .

Graph: The graph for the function $h\left(x\right)=\frac{x-4}{3}$ and its inverse function $h^{-1}\left(x\right)=3x+4$ is shown as,