Functions f and g are such that f(x) = x^2, g(x) = x-3. Solve gf(x)=g^-1(x)

First, we substitute in our functions f and g. We can do this in two ways.1) Find g^-1:As g takes 3 from x, the inverse operation must add 3 to x. So g^-1(x) = x + 3Then our equation gf(x) = g^-1(x) becomes:g(x^2) = x + 3 --> x^2 - 3 = x + 3 --> x^2 - x - 6 = 0, so x = 3 or x = -22) Don't find g^-1:If we apply g to both sides, we get:g^2f(x) = gg^-1(x) --> x^2 - 6 = x, so x = 3 or x = -2
Because g is quite simple in this problem, finding g^-1 is easy, so we can do it either way. But if g was more complicated (g = x^3 - x^2 + 1, say) then finding g^-1 may not be possible, and we may have to do it either way. In maths we often find there are multiple ways of finding the right answer.