Question 198357
I'm assuming that the function is {{{f(x)=-3+sqrt(-9+6x)}}}



{{{f(x)=-3+sqrt(-9+6x)}}} Start with the given function.



Take note that the domain is <font size="8">[</font>*[Tex \LARGE \frac{3}{2},\infty]<font size="8">)</font> and the range is <font size="8">[</font>*[Tex \LARGE -3,\infty]<font size="8">)</font> (graph the function if you are not sure)



{{{y=-3+sqrt(-9+6x)}}} Replace f(x) with y



{{{x=-3+sqrt(-9+6y)}}} Swap each x and y. The goal now is to solve for y to find the inverse.



{{{x+3=sqrt(-9+6y)}}} Add 3 to both sides.



{{{(x+3)^2=(sqrt(-9+6y))^2}}} Square both sides (to undo the square root)



{{{(x+3)^2=-9+6y}}} Square the square root to eliminate it.



{{{x^2+6x+9=-9+6y}}} FOIL



{{{x^2+6x+9+9=6y}}} Add 9 to both sides.



{{{x^2+6x+18=6y}}} Combine like terms.



{{{(x^2+6x+18)/6=y}}} Divide both sides by 6.



{{{y=(x^2+6x+18)/6}}} Rearrange the equation.



So the inverse function is *[Tex \LARGE f^{-1}(x)=\frac{x^2+6x+18}{6}]



Now because the domain and range for f(x) was <font size="8">[</font>*[Tex \LARGE \frac{3}{2},\infty]<font size="8">)</font> and the range is <font size="8">[</font>*[Tex \LARGE -3,\infty]<font size="8">)</font>, this means that the domain and range for the inverse are:



Domain: <font size="8">[</font>*[Tex \LARGE -3,\infty]<font size="8">)</font> 

Range: <font size="8">[</font>*[Tex \LARGE \frac{3}{2},\infty]<font size="8">)</font>