Question

In the following exercises, determine whether or not the given functions are inverses.
$f\left(x\right)=x+8andg\left(x\right)=x-8$

Short answer

Yes, the given functions are inverses of each other.

Step-by-step solution

Step 1. Given and explanation.

We have $f\left(x\right)=x+8andg\left(x\right)=x-8$.

We know that if functions are inverse of each other, so $f\left(g\right(x\left)\right)=xandg\left(f\right(x\left)\right)=x$.

So by verifying this, we will check whether they are inverse of each other or not.

Step 2. Verifying f(g(x))=x

Here, we need to put value of $g\left(x\right)$in place of $x$in .

So this means $f(x-8)$.

Solving it will give us,
$$\begin{gathered} f(x-8)=x \\ \left[(x-8)+8\right]=x \\ x-8+8=x \\ x=x \end{gathered}$$

Hence verified.

Step 3. Verifying g(f(x))=x.

Here, we need to put the value of $f\left(x\right)$in place of $x$in $g\left(x\right)$.

So this means $g(x+8)$.

Solving it will give us,
$$\begin{gathered} g(x+8)=x \\ \left[(x+8)-8\right]=x \\ x+8-8=x \\ x=x \end{gathered}$$
Hence verified.