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.

Answered by William C. Maths tutor

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