Question

Verify that the functions are inverse functions.

$f\left(x\right)=2x+6$

$g\left(x\right)=\frac{x-6}{2}$

Short answer

As

$$\begin{gathered} g\left[f\right(x\left)\right]=x \\ f\left[g\right(x\left)\right]=x \end{gathered}$$the functions are inverses of each other

Step-by-step solution

Given information

Given the functions

$$\begin{gathered} f\left(x\right)=2x+6 \\ g\left(x\right)=\frac{x-6}{2} \end{gathered}$$

Set g[f(x)]=x

Substituting the values, we get

$$\begin{gathered} g(2x+6)=2\left(\frac{x-6}{2}\right)+6 \\ g(2x+6)=x-6+6 \\ g(2x+6)=x \end{gathered}$$

Set f[g(x)]=x

Substituting the values, we get

$$\begin{gathered} f\left(\frac{x-6}{2}\right)=2\left(\frac{x-6}{2}\right)+6 \\ f\left(\frac{x-6}{2}\right)=x-6+6 \\ f\left(\frac{x-6}{2}\right)=x \end{gathered}$$