Function Composition and Decomposition in College Algebra

Function Composition and Decomposition

Evaluating a Function

Evaluating a function involves substituting a specific value for the variable in the function's formula. This process is foundational in algebra and is used to determine the output for a given input.

• Definition: If $f(x)$ is a function, then $f(a)$ is the value of the function when $x = a$.

• Example: For $f(x) = x^2 + 3x - 10$, to evaluate $f(7)$:

$f(7) = (7)^2 + 3(7) - 10 = 49 + 21 - 10 = 60$

Composing a Function

Function composition involves substituting one function into another. The notation $f(g(x))$ means that the output of $g(x)$ becomes the input for $f(x)$.

• Definition: The composition of $f$ and $g$ is written as $f ig( g(x) ig )$ or $(f ext{ o } g)(x)$.

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

$f(g(x)) = (x - 2)^2 + 3(x - 2) - 10$

Notation and Order in Composition

Function composition is often written as $f ig( g(x) ig )$ or $f ext{ o } g(x)$. The first letter is the outside function, and the second letter is the inside function.

• Key Point: $f ext{ o } g(x) = f(g(x))$

• Example: If $f(x) = x + 4$ and $g(x) = x^2 - 3$, then:

(A) $f(g(x)) = (x^2 - 3) + 4 = x^2 + 1$ (B) $g(f(x)) = (x + 4)^2 - 3 = x^2 + 8x + 13$

Practice with Composition

• Example: Given $f(x) = \sqrt{x + 4}$ and $g(x) = (x - 2)^2 - 4$, find:

(A) $f(g(x))$ (B) $g(f(x))$

Evaluating Composed Functions

Sometimes, you need to evaluate a composed function at a specific value. There are two common methods:

• Method 1: Compose then Evaluate

• Method 2: Evaluate Inside then Evaluate Outside

Method 1	Method 2
Find $f(g(x))$ first, then substitute the value for $x$.	Evaluate $g(x)$ at the given value, then use that result as input for $f(x)$.
Example: $f(x) = x + 3$, $g(x) = x - 1$, find $f(g(3))$: $g(3) = 2$, $f(2) = 5$	Example: $f(x) = x^2$, $g(x) = x - 1$, find $f(g(3))$: $g(3) = 2$, $f(2) = 4$

Domain of Composite Functions

To find the domain of a composite function $f(g(x))$, you must consider the domains of both $f$ and $g$.

• Exclude any $x$-values not defined for $g(x)$.

• Exclude any $x$-values that make $g(x)$ not defined for $f$.

• Example: If $f(x) = \frac{1}{x-2}$ and $g(x) = \sqrt{x}$, then $f(g(x))$ is defined for $x > 0$ and $x \neq 4$.

Decomposing Functions

Decomposition is the reverse of composition. It involves expressing a function as the composition of two or more simpler functions.

• Definition: If $h(x) = f(g(x))$, then $h(x)$ is decomposed into $f$ and $g$.

• Example: $h(x) = \sqrt{x - 2}$ can be decomposed as $f(x) = \sqrt{x}$ and $g(x) = x - 2$.

• Example: $h(x) = \frac{1}{x^2 - 3x - 10}$ can be decomposed as $f(x) = \frac{1}{x}$ and $g(x) = x^2 - 3x - 10$.

Summary Table: Function Composition vs. Decomposition

Process	Definition	Example
Composition	Substitute one function into another	$f(g(x))$
Decomposition	Express a function as a composition of two or more functions	$h(x) = f(g(x))$

Key Formulas

• Function Composition: $(f \circ g)(x) = f(g(x))$

• Domain of Composite Function: $\text{Domain of } f(g(x)) = \{ x \mid x \text{ in domain of } g \text{ and } g(x) \text{ in domain of } f \}$

Additional info:

• Function composition is a key concept in College Algebra, especially in understanding how complex functions are built from simpler ones.

• Decomposition helps in analyzing and simplifying functions for further study, such as in calculus or advanced algebra.