Section 2.3: The Composition of Functions – Precalculus Study Notes

Composition of Functions

Definition and Notation

The composition of functions is a fundamental concept in algebra and precalculus, allowing us to combine two functions to form a new function. The composite function f ˆ g, read as "f composed with g," is defined as follows:

• Definition: The composite function f ˆ g is given by:

$(f \circ g)(x) = f(g(x))$

• Here, x must be in the domain of g, and g(x) must be in the domain of f.

Objectives

• Find the composition of two functions and determine the domain of the composition.

• Decompose a function as a composition of two functions.

Examples of Function Composition

Let us consider two functions and compute their compositions:

• Given: $f(x) = 2x - 5$ and $g(x) = x^2 - 3x + 8$

• Find: $(f \circ g)(x)$ and $(g \circ f)(x)$

Solution:

• $(f \circ g)(x) = f(g(x)) = f(x^2 - 3x + 8)$

• $= 2(x^2 - 3x + 8) - 5$

• $= 2x^2 - 6x + 16 - 5$

• $= 2x^2 - 6x + 11$

• Note: The slide shows $2x^2 - 16x - 21$; this appears to be a calculation error. The correct expansion is $2x^2 - 6x + 16 - 5 = 2x^2 - 6x + 11$. Additional info: Always check algebraic expansion carefully.

• $(g \circ f)(x) = g(f(x)) = g(2x - 5)$

• $= (2x - 5)^2 - 3(2x - 5) + 8$

• $= 4x^2 - 20x + 25 - 6x + 15 + 8$

• $= 4x^2 - 26x + 48$

Domain of Composite Functions

When composing functions, the domain of the composite function is determined by the domains of both functions:

• The domain of $f \circ g$ consists of all $x$ in the domain of $g$ such that $g(x)$ is in the domain of $f$.

• The domain of $g \circ f$ consists of all $x$ in the domain of $f$ such that $f(x)$ is in the domain of $g$.

Example: Domain Analysis

• Given: $f(x) = \sqrt{x}$ and $g(x) = x - 3$

• Find: $(f \circ g)(x)$ and $(g \circ f)(x)$

Solution:

• $(f \circ g)(x) = f(g(x)) = \sqrt{x - 3}$

• $(g \circ f)(x) = g(f(x)) = \sqrt{x} - 3$

Domain of $f(x)$: $x \geq 0$ (i.e., $[0, \infty)$)

Domain of $g(x)$: All real numbers $(-\infty, \infty)$

Domain of $f \circ g$: $x - 3 \geq 0 \implies x \geq 3$ (i.e., $[3, \infty)$)

Domain of $g \circ f$: Since $f(x)$ outputs $[0, \infty)$ and $g$ is defined for all real numbers, the domain is $[0, \infty)$.

Decomposing Functions

In calculus and precalculus, it is often useful to express a function as the composition of two simpler functions. This process is called decomposition.

• Example: Given $h(x) = (2x - 3)^5$, find $f(x)$ and $g(x)$ such that $h(x) = (f \circ g)(x)$.

• Let $f(x) = x^5$ and $g(x) = 2x - 3$

• Then $h(x) = f(g(x)) = (2x - 3)^5$

Summary Table: Composition and Domain

Composite Function	Expression	Domain
$f \circ g$	$f(g(x))$	All $x$ in domain of $g$ such that $g(x)$ is in domain of $f$
$g \circ f$	$g(f(x))$	All $x$ in domain of $f$ such that $f(x)$ is in domain of $g$

Key Points

• Composition of functions allows us to build complex functions from simpler ones.

• The order of composition matters: $f \circ g$ is generally not the same as $g \circ f$.

• Always check the domain of the composite function carefully.

• Decomposition is useful for recognizing underlying structure in functions, especially for calculus operations like differentiation and integration.

Additional info: In advanced mathematics, composition is also used in function transformations and in defining inverse functions.