Question

Find f '(x) of f(x) = 3/2x+4

Answer 1

Given:

$f(x)=3\sqrt[]{2x+4}$

The objective is to find inverse of the function.

Consider the given function as,

$y=3\sqrt[]{2x+4}$

The inverse of the function can be calculated by replacing the varibales x and y of the function.

Then, the function will be,

$x=3\sqrt[]{2y+4}$

Now, solve for y.

$\begin{gathered} \frac{x}{3}=\sqrt[]{2y+4} \\ 2y+4=(\frac{x}{3})^2 \\ 2y+4=\frac{x^2}{3^2}^{} \\ 2y=\frac{x^2}{9}^{}-4 \\ y=\frac{1}{2}(\frac{x^2}{9}^{}-4) \\ y=\frac{x^2}{18}^{}-\frac{4}{2} \\ y=\frac{x^2}{18}^{}-2 \end{gathered}$

Thus, the inverse function is,

$f^{-1}(x)=\frac{x^2}{18}^{}-2$

Hence, the required inverse of the function is obtained.