Question

Find the inverse function of the function f(x) = 4x 2/5on the domain x > 0.f-1(x) = x-5/2/4f'(x) =x5/2/4f-'(x) = (x/4)5/4f-1(x) = (x/4)-5/2

Answer 1

Inverse Function

We are given the function

`f(x)=4x^{(2)/(5)}`

To find the inverse function of f, we follow the procedure:

* Make y=f(x)

`y=4x^{(2)/(5)}`

* Solve for x. Take the 5th power

`\begin{gathered} y^5=(4x^{(2)/(5)})^5 \\ \text{Operating:} \\ y^5=4^5x^2 \\ \text{Dividing by }4^5\colon \\ ((y)/(4))^5=x^2 \\ \text{Taking the square root:} \\ x=\sqrt[\square]{((y)/(4))^5}=((y)/(4))^{(5)/(2)} \end{gathered}`

* Swapping the letters:

`y=((x)/(4))^{(5)/(2)}`

Third choice