Given the function
[tex]f(x)=\sqrt[3]{x-15}[/tex]
we want to find the its inverse function
[tex]f^{-1}(x).[/tex]
In order to do that, we will replace f(x) with y. This is done to make the rest of the process easier. So, we have
[tex]y=\sqrt[3]{x-15}[/tex]
Now, let's replace every x with a y and every y with an x,
[tex]x=\sqrt[3]{y-15}[/tex]
and we must solve this equation for y. Then, by raising both sides to the third power, we get
[tex]x^3=y-15[/tex]
and by adding 15 to both sides, we have
[tex]\begin{gathered} x^3+15=y \\ or\text{ equivalently,} \\ y=x^3+15 \end{gathered}[/tex]
Finally, by replacing y with f^-1(x), we have
[tex]f^{-1}(x)=x^3+15[/tex]
Therefore, the answer is option C.
