Express the given function h as a composition of two functions ƒ and g so that h(x) = (fog) (x).
h(x) = (3x − 1)⁴

Verified step by step guidance

1. Step 1: Understand the problem. We are tasked with expressing the given function h(x) = (3x − 1)^4 as a composition of two functions ƒ(x) and g(x), such that h(x) = ƒ(g(x)).
2. Step 2: Identify the inner function g(x). Notice that the expression (3x − 1) is inside the power of 4. This suggests that g(x) = 3x − 1.
3. Step 3: Identify the outer function ƒ(x). The outer operation is raising the input to the power of 4. Therefore, ƒ(x) = x^4.
4. Step 4: Verify the composition. Substitute g(x) into ƒ(x) to check if h(x) = ƒ(g(x)). Substituting g(x) = 3x − 1 into ƒ(x) = x^4 gives ƒ(g(x)) = (3x − 1)^4, which matches h(x).
5. Step 5: Conclude that the functions are ƒ(x) = x^4 and g(x) = 3x − 1, 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 mathematical notation, if we have two functions f(x) and g(x), their composition is denoted as (f o g)(x) = f(g(x)). Understanding this concept is crucial for expressing a function as a composition of two simpler functions.

Identifying Functions

To express a function as a composition, it's essential to identify suitable functions f and g that, when composed, yield the original function h. This often involves recognizing patterns or transformations within the function. For example, in h(x) = (3x - 1)^4, one might consider g(x) = 3x - 1 and f(x) = x^4 to facilitate the composition.

Polynomial Functions

Polynomial functions are expressions that involve variables raised to whole number powers, combined using addition, subtraction, and multiplication. The function h(x) = (3x - 1)^4 is a polynomial function, specifically a quartic function. Understanding the properties of polynomial functions, such as their behavior and transformations, is essential for manipulating and composing them effectively.