Function Composition and Operations in College Algebra

Function Composition

Definition and Notation

Function composition is a fundamental concept in algebra where two functions are combined to form a new function. The composition of functions f and g, denoted as (f ˆ g)(x), means applying g first and then applying f to the result of g(x).

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

• Domain: $x$ must be in the domain of $g$, and $g(x)$ must be in the domain of $f$.

Example: If $f(x) = x^2 - 3x$ and $g(x) = 4x - 1$, then:

• $(f \circ g)(x) = f(g(x)) = (4x - 1)^2 - 3(4x - 1)$

Evaluating Compositions at Specific Values

To evaluate a composition at a specific value, substitute the value into the inner function and then apply the outer function.

• Example: $(f \circ g)(3) = f(g(3))$

• First, find $g(3)$, then substitute that result into $f$.

Domain of Composed Functions

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

• Check the domain of $g$.

• Check that $g(x)$ outputs values in the domain of $f$.

Function Operations

Combining Functions

Functions can be combined using addition, subtraction, multiplication, and division, as well as composition.

• Addition: $(f + g)(x) = f(x) + g(x)$

• Subtraction: $(f - g)(x) = f(x) - g(x)$

• Multiplication: $(f \cdot g)(x) = f(x) \cdot g(x)$

• Division: $(\frac{f}{g})(x) = \frac{f(x)}{g(x)}$, $g(x) \neq 0$

Examples of Function Operations

• Given: $k(x) = \frac{1}{\sqrt{2x-5}}$

• Domain: $2x-5 > 0 \implies x > 2.5$

• Given: $h(x) = 2 + (3x + 1)^4$

• Domain: All real numbers (since any real $x$ can be used in a polynomial and addition).

Summary Table: Function Composition and Operations

Operation	Notation	Formula	Domain Considerations
Composition	$(f \circ g)(x)$	$f(g(x))$	$x$ in domain of $g$ and $g(x)$ in domain of $f$
Addition	$(f + g)(x)$	$f(x) + g(x)$	Intersection of domains of $f$ and $g$
Subtraction	$(f - g)(x)$	$f(x) - g(x)$	Intersection of domains of $f$ and $g$
Multiplication	$(f \cdot g)(x)$	$f(x) \cdot g(x)$	Intersection of domains of $f$ and $g$
Division	$(\frac{f}{g})(x)$	$\frac{f(x)}{g(x)}$	Intersection of domains of $f$ and $g$, $g(x) \neq 0$

Additional info:

• When composing functions, always pay attention to the order: $(f \circ g)(x) \neq (g \circ f)(x)$ in general.

• For radical and rational functions, domain restrictions are especially important.