In Exercises 89–90, express the given function h as a composition of two functions f and g so that h(x) = (f ○ g)(x). h(x) = (x^2 + 2x - 1)^4

Verified step by step guidance

1. Identify the inner function and the outer function in the given expression. Here, the inner function can be the expression inside the parentheses, and the outer function can be the operation applied to this expression.
2. Let the inner function be \( g(x) = x^2 + 2x - 1 \). This represents the expression inside the parentheses.
3. Let the outer function be \( f(u) = u^4 \). This represents raising the expression to the fourth power.
4. Express the original function \( h(x) \) as a composition of \( f \) and \( g \) using the notation \( (f \circ g)(x) = f(g(x)) \).
5. Substitute \( g(x) \) into \( f(u) \) to verify: \( f(g(x)) = (g(x))^4 = (x^2 + 2x - 1)^4 \), which matches \( h(x) \).

Key Concepts

Function Composition

Function composition involves combining two functions, where the output of one function becomes the input of another. This is denoted as (f ○ g)(x) = f(g(x)). Understanding how to decompose a function into two simpler functions is essential for solving problems that require expressing a function as a composition.

Polynomial Functions

A polynomial function is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients. In the given function h(x) = (x^2 + 2x - 1)^4, the inner function g(x) can be identified as the polynomial x^2 + 2x - 1, which is raised to the fourth power by the outer function f. Recognizing polynomial structures is crucial for effective manipulation and composition.

Exponentiation

Exponentiation is a mathematical operation involving a base raised to a power, indicating how many times to multiply the base by itself. In the context of the function h(x), the outer function f is defined as f(u) = u^4, where u represents the output of the inner function g. Understanding exponentiation is vital for accurately expressing and simplifying functions in algebra.