Question

What is the inverse of f if f(x) = 3 sqrt (x-5) (see image)

Answer 1

The given function is:

`f(x)=\sqrt[3]{x-5}`

To find the inverse set f(x)=y

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

Now, solve for x:

cube both sides of the equation:

`\begin{gathered} y^3=(\sqrt[3]{x-5})^3 \\ \text{Simplify} \\ y^3=x-5 \end{gathered}`

Add 5 to both sides:

`\begin{gathered} y^3+5=x-5+5 \\ y^3+5=x \end{gathered}`

Now, switch x and y and reorder terms:

`\begin{gathered} x^3+5=y \\ y=x^3+5 \end{gathered}`

Replace y by f^-1(x):

`f^(-1)(x)=x^3+5`

This is the inverse of the function.