Question 163985
{{{f(x)=-sqrt(x+2)+3}}} Start with the given function



{{{y=-sqrt(x+2)+3}}} Replace f(x) with y



{{{x=-sqrt(y+2)+3}}} Switch x and y



The goal is now to solve for y:


{{{x-3=-sqrt(y+2)}}} Subtract 3 from both sides



{{{(x-3)^2=y+2}}} Square both sides



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



{{{x^2-6x+7=y+2}}} FOIL the left side



{{{x^2-6x+7=y}}} Combine like terms.



So the solution is {{{y=x^2-6x+7}}} which means that the inverse function is {{{f^-1(x)=x^2-6x+7}}} where the domain of {{{f^-1(x)}}} is {{{x<=3}}} and the range is {{{y>=-2}}}