Find f(g(x)) and g (f(x)) and determine whether each pair of functions ƒ and g are inverses of each other. f(x) = ∛(x − 4) and g(x) = x³ +4

Verified step by step guidance

1. First, recall that the composition of functions f(g(x)) means substituting g(x) into f(x). So, write down f(g(x)) as f(g(x)) = \sqrt[3]{g(x) - 4}.
2. Next, substitute g(x) = x^3 + 4 into the expression for f(g(x)), giving f(g(x)) = \sqrt[3]{(x^3 + 4) - 4}.
3. Simplify the expression inside the cube root: (x^3 + 4) - 4 simplifies to x^3, so f(g(x)) = \sqrt[3]{x^3}.
4. Now, recall that the cube root of x cubed is just x, so f(g(x)) simplifies to x.
5. Next, find g(f(x)) by substituting f(x) into g(x). Write g(f(x)) = (f(x))^3 + 4, then substitute f(x) = \sqrt[3]{x - 4} to get g(f(x)) = (\sqrt[3]{x - 4})^3 + 4. Simplify this expression to check if it equals x.

Key Concepts

Function Composition

Function composition involves applying one function to the result of another, denoted as f(g(x)) or g(f(x)). It requires substituting the entire expression of one function into the variable of the other, allowing us to analyze how functions combine and transform inputs.

Inverse Functions

Inverse functions reverse the effect of each other, meaning f(g(x)) = x and g(f(x)) = x for all x in the domains. Identifying inverses involves checking if composing the functions in both orders returns the original input, confirming they undo each other's operations.

Cube Roots and Cubes

The cube root function, ∛x, and the cube function, x³, are inverse operations. Understanding how these functions interact, especially with shifts like (x - 4) or +4, is essential to correctly compose and verify if two functions are inverses.