Question

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

$f\left(x\right)=9x$

Short answer

The inverse of the function$f\left(x\right)=9x$is

localid="1644857079934" $f^{-1}\left(x\right)=\frac{x}{9}$

Step-by-step solution

Step 1. Given data

The given function is$f\left(x\right)=9x$

Step 2. Substitute y for f(x)

Replace$f\left(x\right)$with $y$

$$\begin{gathered} f\left(x\right)=9x \\ y=9x \end{gathered}$$

Step 3. Exchange x & y

Replace $x$with $y$and $y$with $x$

$$\begin{gathered} y=9x \\ x=9y \end{gathered}$$

Step 4. solve the equation for y

Divide both sides of the equation by $9$

$$\begin{gathered} \frac{x}{9}=\frac{9y}{9} \\ \frac{x}{9}=y \end{gathered}$$

Step 5. Substitute f-1(x) for y

Replace $y$with $f^{-1}\left(x\right)$

$$\begin{gathered} \frac{x}{9}=y \\ \frac{x}{9}=f^{-1}\left(x\right) \\ {f}^{-1}\left(x\right)=\frac{x}{9} \end{gathered}$$

Step 6. verification

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

$$\begin{gathered} f^{-1}\left(f\right(x\left)\right)\stackrel{?}{=}x \\ {f}^{-1}\left(9x\right)\stackrel{?}{=}x \\ \frac{9x}{9}\stackrel{?}{=}x \\ x=x \end{gathered}$$

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

$$\begin{gathered} f\left(f^{-1}\right(x\left)\right)\stackrel{?}{=}x \\ f\left(\frac{x}{9}\right)\stackrel{?}{=}x \\ 9\left(\frac{x}{9}\right)\stackrel{?}{=}x \\ x=x \end{gathered}$$