Question

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

$f\left(x\right)=-5x-4$

Short answer

The inverse of the function is$f^{-1}\left(x\right)=\frac{-x-4}{5}$

Step-by-step solution

Step 1. Given data

The given function is$f\left(x\right)=-5x-4$

Step 2. Substitute y for f(x)

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

role="math" localid="1644861619303" $$\begin{gathered} f\left(x\right)=-5x-4 \\ y=-5x-4 \end{gathered}$$

Step 3. Exchange x & y

Interchange variables $x\&y$

role="math" localid="1644863097117" $$\begin{gathered} y=-5x-4 \\ x=-5y-4 \end{gathered}$$

Step 4. solve the equation for y

Add $4$to both sides

$$\begin{gathered} x+4=-5y-4+4 \\ x+4=-5y \end{gathered}$$

Divide both sides by $-5$

role="math" localid="1644864455953" $$\begin{gathered} \frac{x+4}{-5}=\frac{-5y}{-5} \\ \frac{-x-4}{5}=y \end{gathered}$$

Step 5. Substitute f-1(x) for y

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

role="math" localid="1644866690115" $$\begin{gathered} \frac{-x-4}{5}=y \\ \frac{-x-4}{5}=f^{-1}\left(x\right) \\ {f}^{-1}\left(x\right)=\frac{-x-4}{5} \end{gathered}$$

Step 6. verification

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

localid="1644866587793" $$\begin{gathered} f^{-1}(-5x-4)\stackrel{?}{=}x \\ \frac{-(-5x-4)-4}{5}\stackrel{?}{=}x \\ \frac{5x+4-4}{5}\stackrel{?}{=}x \\ x=x \end{gathered}$$

Show localid="1644867863016" $f\left(f^{-1}\right(x\left)\right)=x$

$$\begin{gathered} f\left(\frac{-x-4}{5}\right)\stackrel{?}{=}x \\ -5\left(\frac{-x-4}{5}\right)-4\stackrel{?}{=}x \\ x+4-4\stackrel{?}{=}x \\ x=x \end{gathered}$$