Question

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

$f\left(x\right)=\sqrt[5]{-4x-3}$

Short answer

The inverse of the given function is$f^{-1}\left(x\right)=\frac{-x^5-3}{4}$

Step-by-step solution

Step 1. Given information    

The given function is$\sqrt[5]{-4x-3}$

Step 2. Interchange the variable  x and y.

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

$y=\sqrt[5]{-4x-3}$

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

role="math" localid="1644936553145" $x=\sqrt[5]{-4y-3}$

Step 3. solve for  y

Solving for $y$, we get:

role="math" localid="1644938527446" $$\begin{gathered} x=\sqrt[5]{-4y-3} \\ {x}^5=-4y-3 \\ -4y=x^5+3 \\ y=\frac{-x^5-3}{4} \end{gathered}$$

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

$f^{-1}\left(x\right)=\frac{-x^5-3}{4}$

Step 4. Verify that the functions are inverses. 

Substituting $f\left(x\right)$in role="math" localid="1644938933535" $f^{-1}\left(f\left(x\right)\right)=x$, we get:

$$\begin{gathered} f^{-1}\left(f\left(x\right)\right)=x \\ {f}^{-1}\left(\sqrt[5]{-4x-3}\right)=x \\ \sqrt[5]{-4\left(\frac{-x^5-3}{4}\right)-3}=x \\ \sqrt[5]{x^5-3+3}=x \\ \sqrt[5]{x^5}=x \\ x=x \end{gathered}$$

This is true.

Substituting $f^{-1}\left(x\right)$in role="math" localid="1644940940489" $f\left(f^{-1}\left(x\right)\right)=x$, we get:

$$\begin{gathered} f\left(f^{-1}\left(x\right)\right)=x \\ f\left(\frac{-x^5-3}{4}\right)=x \\ \sqrt[5]{-4·\left(\frac{-x^5-3}{4}\right)-3} \\ \sqrt[5]{x^5+3-3}=x \\ \sqrt[5]{x^5}=x \\ x=x \end{gathered}$$

This is also true.

Thus, the obtained inverse is correct.