Working with composite functions
Find possible choices for outer and inner functions ƒ and g such that the given function h equals ƒ o g.
h(x) = (x³ - 5)¹⁰

Verified step by step guidance

1. Step 1: Identify the structure of the composite function. The given function is \( h(x) = (x^3 - 5)^{10} \). This suggests that there is an inner function and an outer function involved.
2. Step 2: Determine the inner function \( g(x) \). Look for a function inside another function. Here, \( g(x) = x^3 - 5 \) is a natural choice for the inner function because it is the expression inside the power.
3. Step 3: Determine the outer function \( f(u) \). The outer function is applied to the result of the inner function. In this case, \( f(u) = u^{10} \) is the outer function, where \( u = g(x) = x^3 - 5 \).
4. Step 4: Verify the composition. Check that \( f(g(x)) = f(x^3 - 5) = (x^3 - 5)^{10} \) matches the original function \( h(x) \).
5. Step 5: Conclude the choices. The functions \( f(u) = u^{10} \) and \( g(x) = x^3 - 5 \) are valid choices for the outer and inner functions, respectively, such that \( h(x) = f(g(x)) \).

Key Concepts

Composite Functions

A composite function is formed when one function is applied to the result of another function. It is denoted as (ƒ o g)(x) = ƒ(g(x)), where g is the inner function and ƒ is the outer function. Understanding how to decompose a function into its components is essential for solving problems involving composite functions.

Function Notation

Function notation is a way to represent functions and their relationships. In this context, h(x) represents the output of the function h for a given input x. Recognizing how to manipulate and interpret function notation is crucial for identifying the inner and outer functions in a composite function.

Power Functions

Power functions are functions of the form f(x) = x^n, where n is a real number. In the given function h(x) = (x³ - 5)¹⁰, the outer function can be identified as a power function, while the inner function can be the expression inside the parentheses. Understanding power functions helps in determining how to break down the composite function into its components.