Question

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

$f\left(x\right)=\frac{x}{6}$

Short answer

The inverse of the function is$f^{-1}\left(x\right)=6x$

Step-by-step solution

Step 1. Given data

The given function is$f\left(x\right)=\frac{x}{6}$

Step 2. Substitute y for f(x)

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

role="math" localid="1644861377311" $$\begin{gathered} f\left(x\right)=\frac{x}{6} \\ y=\frac{x}{6} \end{gathered}$$

Step 3. Exchange x & y

Interchange variables $x\&y$

$$\begin{gathered} y=\frac{x}{6} \\ x=\frac{y}{6} \end{gathered}$$

Step 4. solve the equation for y

Multiply both sides by $6$

role="math" localid="1644864113883" $$\begin{gathered} x\left(6\right)=\frac{y}{6}\left(6\right) \\ 6x=y \end{gathered}$$

Step 5. Substitute f-1(x) for y

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

role="math" localid="1644865216304" $$\begin{gathered} 6x=f^{-1}\left(x\right) \\ {f}^{-1}\left(x\right)=6x \end{gathered}$$

Step 6. verification

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

role="math" localid="1644865303824" $$\begin{gathered} f^{-1}\left(\frac{x}{6}\right)\stackrel{?}{=}x \\ 6\left(\frac{x}{6}\right)\stackrel{?}{=}x \\ x=x \end{gathered}$$

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

role="math" localid="1644865374407" $$\begin{gathered} f\left(6x\right)\stackrel{?}{=}x \\ \frac{\left(6x\right)}{6}\stackrel{?}{=}x \\ x=x \end{gathered}$$