Question

Find the inverse of the following function:

Answer 1

First, we are going to find the inverse function of $f(x)$ by replacing $f(x)$ with $y$, and then, interchanging $x$ and $y$ and solving for $x$:
$y=-2 \sqrt{3x-1} -5$
$x=-2 \sqrt{3y-1} -5$
$-2 \sqrt{3y-1} =x+5$
$\sqrt{3y-1} = \frac{x+5}{-2}$
$3y-1=( \frac{x+5}{-2} )^{2}$
$3y=( \frac{x+5}{-2} )^{2} +1$
$y= \frac{( \frac{x+5}{-2})^{2}+1 }{3}$

Next we are going to evaluate our inverse function at $x= \frac{1}{3}$:
$y= \frac{( \frac{ \frac{1}{3}+5 }{-2})^{2}+1 }{3}$
$y= \frac{( \frac{ \frac{16}{3} }{-2})^{2}+1 }{3}$
$y= \frac{(- \frac{8}{3})^{2}+1 }{3}$
$y= \frac{ \frac{64}{9}+1 }{3}$
$y= \frac{ \frac{73}{9} }{3}$
$y= \frac{73}{27}$

We can conclude that the inverse function of $f(x)=-2 \sqrt{3x-1} -5$ when $x= \frac{1}{3}$ is $\frac{73}{27}$