Function Composition in Precalculus: Concepts, Evaluation, Domain, and Decomposition

Function Composition

Introduction to Function Composition

Function composition is a fundamental concept in precalculus, where the output of one function becomes the input of another. This process allows for the creation of new functions and is essential for understanding more advanced mathematical topics.

• Definition: The composition of functions f and g, written as $(f \circ g)(x)$ or $f(g(x))$, means applying g first, then f to the result.

• Notation: $(f \circ g)(x) = f(g(x))$

• Order: The function closest to x is applied first.

Evaluating a Function vs. Composing a Function

Evaluating a Function

To evaluate a function, substitute a specific value for the variable and simplify.

• Example: If $f(x) = x^2 + 3x - 10$, then $f(2) = 2^2 + 3(2) - 10 = 4 + 6 - 10 = 0$

Composing a Function

To compose functions, substitute one function into another.

• Example: If $f(x) = x^2 + 3x - 10$ and $g(x) = x + 2$, then $f(g(x)) = (x + 2)^2 + 3(x + 2) - 10$

• Simplify: $(x + 2)^2 + 3(x + 2) - 10 = x^2 + 4x + 4 + 3x + 6 - 10 = x^2 + 7x$

Methods for Evaluating Composite Functions

Method 1: Compose then Evaluate

First, compose the functions, then substitute the value.

• Example: $f(x) = x^2 + 2x + 1$, $g(x) = 2x - 4$, find $f(g(2))$

• Step 1: $g(2) = 2(2) - 4 = 0$

• Step 2: $f(0) = 0^2 + 2(0) + 1 = 1$

Method 2: Evaluate Inside, then Outside

Evaluate the inner function first, then use its result in the outer function.

• Example: $f(x) = x^2 + 2x + 1$, $g(x) = 2x - 4$, find $f(g(2))$

• Step 1: $g(2) = 0$

• Step 2: $f(0) = 1$

Domain of Composite Functions

Finding the Domain

To determine the domain of a composite function $f(g(x))$, follow these steps:

• 1. Find x-values for which $g(x)$ is defined.

• 2. Find x-values for which $f(g(x))$ is defined (i.e., the output of $g(x)$ must be in the domain of $f$).

Example: $f(x) = \sqrt{x}$, $g(x) = x^2 - 4$, find the domain of $f(g(x))$.

• $f(g(x)) = \sqrt{x^2 - 4}$

• Require $x^2 - 4 \geq 0 \implies x \leq -2$ or $x \geq 2$

• Domain: $(-\infty, -2] \cup [2, \infty)$

Decomposing Functions

Reverse of Composition

Decomposing a function involves expressing a given function as the composition of two or more simpler functions.

• Example: $h(x) = \sqrt{2x + 3}$ can be written as $h(x) = f(g(x))$ where $g(x) = 2x + 3$ and $f(x) = \sqrt{x}$

• There are often multiple correct ways to decompose a function.

Practice Problems and Examples

Practice: Composing and Evaluating Functions

• Given $f(x) = x^2 + 4$ and $g(x) = x - 2$, find $f(g(x))$ and $g(f(x))$

• $f(g(x)) = (x - 2)^2 + 4 = x^2 - 4x + 4 + 4 = x^2 - 4x + 8$

• $g(f(x)) = (x^2 + 4) - 2 = x^2 + 2$

Practice: Domain of Composite Functions

• Given $f(x) = x^2$ and $g(x) = \sqrt{x - 2}$, find $f(g(x))$ and its domain.

• $f(g(x)) = (\sqrt{x - 2})^2 = x - 2$

• Domain: $x - 2 \geq 0 \implies x \geq 2$

Practice: Decomposing Functions

• Express $h(x) = \ln(2x + 3)$ as a composition of two functions.

• Let $g(x) = 2x + 3$, $f(x) = \ln(x)$, so $h(x) = f(g(x))$

Summary Table: Function Composition Concepts

Concept	Definition	Example
Evaluating a Function	Substitute a value for x	$f(2)$ for $f(x) = x^2 + 3x - 10$
Composing Functions	Substitute one function into another	$f(g(x))$
Domain of Composite	Values of x for which both functions are defined	$f(g(x))$ domain
Decomposing Functions	Express as composition of simpler functions	$h(x) = f(g(x))$