Chapter 7.2, Problem 24PPS

To determine

The inverse of the function $h\left(x\right)=x^2+4$ and graph the function and its inverse.

Answer to Problem 24PPS

The inverse of function $h\left(x\right)=x^2+4$ is $h^{-1}\left(x\right)=\pm \sqrt{x-4}$ .

Explanation of Solution

Given information: The given function is,

  $h\left(x\right)=x^2+4$

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 $h\left(x\right)=x^2+4$ , the inverse is evaluated as,

  $\begin{array}{c}y=x^2+\mathrm{4......}\left(\text{Plugging }h\left(x\right)=y\right)\\ \text{or, }x=y^2+\mathrm{4......}\left(\text{Exchanging variables}\right)\\ y=\pm \sqrt{x-4}\mathrm{......}\left(\text{Solving for variable }\left(y\right)\right)\\ h^{-1}\left(x\right)=\pm \sqrt{x-4}\mathrm{......}\left(\text{Plugging }y=h^{-1}\left(x\right)\right)\end{array}$

Hence, the inverse of function $h\left(x\right)=x^2+4$ is $h^{-1}\left(x\right)=\pm \sqrt{x-4}$ .

Graph: The graph for the function $h\left(x\right)=x^2+4$ and its inverse function $h^{-1}\left(x\right)=\pm \sqrt{x-4}$ is shown as,