find the inverse of the following function:[tex]f(x)=-2 \sqrt{3x-1} -5 [/tex]if x \geq 1/3[/tex]

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

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

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