Question

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

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

Short answer

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

Step-by-step solution

Step 1. Given information   

The given function is$f\left(x\right)=\sqrt[3]{x-3}$

Step 2. Interchange the variable  x and y

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

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

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

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

Step 3. Solve for y

Solving for $y$, we get:

$$\begin{gathered} x=\sqrt[3]{y-3} \\ {x}^3=y-3 \\ y=x^3+3 \end{gathered}$$

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

$f^{-1}\left(x\right)=x^3+3$

Step 4. Verify that the functions are inverses. 

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

$$\begin{gathered} f^{-1}\left(f\left(x\right)\right)=x \\ {f}^{-1}\left(\sqrt[3]{x-3}\right)=x \\ {\left(\sqrt[3]{x-3}\right)}^2+3=x \\ x-3+3=x \\ x=x \end{gathered}$$

This is True

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

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

This is true.

Thus, the obtained inverse is correct