Question

given function G(x)=1/3(x-2)^2-3 2<x<5 sketch a graph of function y=g^-x and find the inverse function g^-1

Answer 1

to find the inverse of G(x)

let y = 1/3(x-2)^2 - 3

to take the inverse...
Step 1, interchange the x and y variables
Step 2, solve this for y

---> x = 1/3(y-2^2 - 3
3x = (y-2)^2 - 9
(y-2)^2 = 3x+9
y-2 = ±√(3x+9)
y = 2 ± √(3x+9)

so g^-1(x) = 2 ± √(3x+9)

I am not quite sure what you mean by y=g^-x

I think my reply also ties in with your other post,
re domain and range
the domain of a function becomes the range of its inverse, and vice versa

so find g(2) and g(5) for the first function and those values would then be the domain of your inverse