Express the given function h as a composition of two functions ƒ and g so that h(x) = (fog) (x). h(x) = ∛(x² – 9)

Verified step by step guidance

1. Step 1: Understand the problem. The goal is to express the given function h(x) = ∛(x² – 9) as a composition of two functions ƒ(x) and g(x) such that h(x) = (ƒ ∘ g)(x), which means h(x) = ƒ(g(x)).
2. Step 2: Identify the inner function g(x). Look at the expression inside the cube root, x² – 9. This suggests that g(x) = x² – 9.
3. Step 3: Identify the outer function ƒ(x). The outer function operates on the result of g(x). Since h(x) = ∛(x² – 9), the cube root operation applies to g(x). Therefore, ƒ(x) = ∛x or equivalently ƒ(x) = x^(1/3).
4. Step 4: Verify the composition. Substitute g(x) into ƒ(g(x)) to ensure it matches h(x). ƒ(g(x)) = ƒ(x² – 9) = ∛(x² – 9), which is the original h(x).
5. Step 5: Conclude that the functions are ƒ(x) = x^(1/3) and g(x) = x² – 9, and their composition satisfies h(x) = (ƒ ∘ g)(x).

Key Concepts

Function Composition

Function composition involves combining two functions, where the output of one function becomes the input of another. In this context, if h(x) = (f o g)(x), it means h(x) can be expressed as f(g(x)). Understanding how to break down a function into simpler components is essential for solving the problem.

Cube Root Function

The cube root function, denoted as ∛x, is the inverse of the cubic function x³. It is important to recognize how this function behaves, particularly its domain and range, as well as how it can be manipulated algebraically. In the given function h(x) = ∛(x² - 9), understanding the cube root will help in identifying suitable functions f and g.

Quadratic Functions

Quadratic functions are polynomial functions of the form ax² + bx + c, where a, b, and c are constants. In the expression x² - 9, we see a difference of squares, which can be factored. Recognizing the structure of quadratic expressions is crucial for determining how to express h(x) as a composition of two simpler functions.