Composite functions and notation
Let ƒ(x)= x² - 4 , g(x) = x³ and F(x) = 1/(x-3).
Simplify or evaluate the following expressions.
g(ƒ(u))

Verified step by step guidance

Identify the inner function and the outer function in the composite function g(ƒ(u)).

The inner function is ƒ(u) = u^2 - 4.

The outer function is g(x) = x^3.

Substitute the inner function ƒ(u) into the outer function g(x).

Evaluate g(ƒ(u)) by replacing x in g(x) with ƒ(u), resulting in (u^2 - 4)^3.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Composite Functions

A composite function is formed when one function is applied to the result of another function. In mathematical notation, if you have two functions f(x) and g(x), the composite function g(f(x)) means you first apply f to x and then apply g to the result of f. Understanding how to combine functions is essential for simplifying expressions involving multiple functions.

Function Notation

Function notation is a way to denote functions and their operations clearly. For example, f(x) represents a function f evaluated at x. This notation helps in understanding how to manipulate and evaluate functions, especially when dealing with compositions like g(f(u)). Recognizing the roles of different variables and functions in expressions is crucial for accurate calculations.

Simplification of Expressions

Simplification involves reducing an expression to its simplest form, making it easier to work with. This can include combining like terms, factoring, or substituting values. In the context of composite functions, simplifying g(f(u)) requires substituting the expression for f(u) into g and then performing any necessary algebraic operations to arrive at a more manageable form.

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