If the function is expressed in the form f of g

Question

If the function h(x)=(x+2)^8 is expressed in the form f∘g with f(x)=x^8, then find the function g(x).

g(x)= 

Answer 1

Looking at h(x) = (x + 2)⁸ , and f(x) = x⁸ , we can see that g(x) = x + 2 works, since

f o g (x) = f(g(x)) = f(x + 2) = (x + 2)⁸.

If it's not clear how to get to g(x) = x + 2, just note that f has a partial inverse f^-1(y) = y^1/8 , which is the inverse of f(x) when we restrict the variable x to x ≥ 0. In this case, when x + 2 ≥ 0, we have:

f^-1(f(g(x))) = g(x) , since the f and the f^-1 cancel out. We also have:

f^-1(f(g(x))) = (f(g(x)))^1/8 = ((x+2)⁸)^1/8 = (x + 2) , since x + 2 ≥ 0. Note that, in general (y⁸)^1/8 = |y|.

The reasoning above only works for x+2 ≥ 0, since only in this case can we use f^-1 as we had defined, but we can set g(x) = x + 2 and check that it works even if x + 2 < 0.