Express the given function h as a composition of two functions f and g so that h(x) = (f ○ g)(x). h(x) = (x² + 2x - 1)⁴

Verified step by step guidance

1. Identify the outer function and the inner function in the expression \(h(x) = (x^2 + 2x - 1)^4\). The outer function is the one applied last, which in this case is raising to the 4th power.
2. Define the inner function \(g(x)\) as the expression inside the parentheses: \(g(x) = x^2 + 2x - 1\).
3. Define the outer function \(f(x)\) as the function that raises its input to the 4th power: \(f(x) = x^4\).
4. Express \(h(x)\) as a composition of \(f\) and \(g\): \(h(x) = (f \circ g)(x) = f(g(x))\).
5. Verify by substituting \(g(x)\) into \(f\): \(f(g(x)) = (x^2 + 2x - 1)^4\), which matches the original function \(h(x)\).

Key Concepts

Function Composition

Function composition involves combining two functions where the output of one function becomes the input of another. If h(x) = (f ○ g)(x), then h(x) = f(g(x)). Understanding this helps in breaking down complex functions into simpler parts.

Identifying Inner and Outer Functions

To express a function as a composition, identify an inner function g(x) that is substituted into an outer function f(x). For h(x) = (x^2 + 2x - 1)^4, the inner function is the expression inside the parentheses, and the outer function raises that result to the fourth power.

Polynomial Functions

Polynomial functions are expressions involving variables raised to whole-number exponents combined using addition, subtraction, and multiplication. Recognizing polynomial structure helps in selecting appropriate f and g functions for composition.