Question

In the following exercise, find the inverse of each function.

$f\left(x\right)=\sqrt{x+8},x\ge -8$

Short answer

The inverse of the given function is$f^{-1}\left(x\right)=x^2-8$

Step-by-step solution

Step 1. Given information   

The given function is$f\left(x\right)=\sqrt{x+8}$

Step 2. Interchange the variable y and x

Substituting $f\left(x\right)=y$, we get:

$y=\sqrt{x+8}$

interchanging the variable $x$and $y$, we get:

$x=\sqrt{y+8}$

Step 3.  Solve for y

Solving for $y$, we get:

$$\begin{gathered} x=\sqrt{y+8} \\ {x}^2=y+8 \\ y=x^2-8 \end{gathered}$$

Substituting $y=f^{-1}\left(x\right)$, we get:

$f^{-1}\left(x\right)=x^2-8$

Step 4. verify that the functions are inverses.  

Substituting $f\left(x\right)$in $f^{-1}\left(f\left(x\right)\right)=x$, we get:

$$\begin{gathered} f^{-1}\left(f(x\right))=x \\ {\left(\sqrt{x-8}\right)}^2+8=x \\ x-8+8=x \\ x=x \end{gathered}$$

This is true

Substituting $f^{-1\text{'}}\left(x\right)$in $f\left(f^{-1}\left(x\right)\right)=x$

$$\begin{gathered} f\left(f^{-1}\left(x\right)\right)=x \\ f\left(x^2-8\right)=x \\ \sqrt{x^2-8+8}=x \\ x=x \end{gathered}$$

This is true,

Thus, the obtained inverse is correct.