The given function is:
[tex]f(x)=\sqrt[3]{x-5}[/tex]
To find the inverse set f(x)=y
[tex]y=\sqrt[3]{x-5}[/tex]
Now, solve for x:
cube both sides of the equation:
[tex]\begin{gathered} y^3=(\sqrt[3]{x-5})^3 \\ \text{Simplify} \\ y^3=x-5 \end{gathered}[/tex]
Add 5 to both sides:
[tex]\begin{gathered} y^3+5=x-5+5 \\ y^3+5=x \end{gathered}[/tex]
Now, switch x and y and reorder terms:
[tex]\begin{gathered} x^3+5=y \\ y=x^3+5 \end{gathered}[/tex]
Replace y by f^-1(x):
[tex]f^{-1}(x)=x^3+5[/tex]
This is the inverse of the function.
