Question

Confirm that f and g are inverses by showing that f(g(x)) = x and g(f(x)) = x.

f(x) = x3 + 4 and g(x) = Cube root of quantity x minus four.

Answer 1

\[\large\rm f(\color{orangered}{x})=(\color{orangered}{x})^3+4\]Let's replace our x with g(x).
We're going to plug the entire g(x) function into f(x).\[\large\rm f(\color{orangered}{g(x)})=(\color{orangered}{g(x)})^3+4\]On the right side, we'll replace g(x) which what it really is:\[\large\rm f(\color{orangered}{g(x)})=(\color{orangered}{\sqrt[3]{x-4}})^3+4\]

Answer 2

Recall that `cube power` and `cube root` are inverse functions, so they "undo" one another.\[\large\rm f(\color{orangered}{g(x)})=(x-4)+4\]So does f(g(x)) simplify to x?