Question

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

$f\left(x\right)=\frac{1}{x+2}$

Short answer

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

Step-by-step solution

Step 1. Given information 

The given function is$f\left(x\right)=\frac{1}{x+2}$

Step 2.  Interchange the variable y and x

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

$y=\frac{1}{x+2}$

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

$x=\frac{1}{y+2}$

Step 3. Solve for y

Solving for $y$, we get:

$$\begin{gathered} x=\frac{1}{y+2} \\ y+2=\frac{1}{x} \\ y=\frac{1}{x}-2 \end{gathered}$$

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

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

Step 4. Verify that the functions are inverses. 

Substituting $f\left(x\right)$in localid="1644864402589" $f^{-1}\left(f\right(x\left)\right)=x$, we get:

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

This is true.

Substituting $f^{-1}\left(x\right)$in localid="1644864638856" $f\left(f^{-1}\left(x\right)\right)=x$, we get:

localid="1644864863049" $$\begin{gathered} f\left(f^{-1}\left(x\right)\right)=x \\ f\left(\frac{1}{x}-2\right)=x \\ \frac{1}{{\displaystyle \frac{1}{x}}-2+2}=x \\ \frac{1}{{\displaystyle \frac{1}{x}}}=x \\ x=x \end{gathered}$$

The obtained inverse is correct.