Combining Functions: Sums, Products, Quotients, and Composition

2.2 Combining Functions

Overview

This section explores methods for combining functions in College Algebra, including sums, differences, products, quotients, and composition. Understanding these operations is essential for constructing new functions and analyzing their domains.

Sums, Differences, Products, and Quotients of Functions

Definitions and Domain Rules

Given two functions f and g with domains A and B, their combinations are defined as follows:

• Sum: $(f+g)(x) = f(x) + g(x)$ Domain: $A igcap B$

• Difference: $(f-g)(x) = f(x) - g(x)$ Domain: $A igcap B$

• Product: $(fg)(x) = f(x)g(x)$ Domain: $A igcap B$

• Quotient: $igg( rac{f}{g}igg)(x) = rac{f(x)}{g(x)}$ Domain: $ig\{ x \\in A igcap B \\mid g(x) eq 0 ig\\}$

Example: Domain Calculation

Suppose $f(x) = \frac{1}{x-2}$ and $g(x) = \sqrt{x}$. The domain of $f$ is $\{x \mid x \neq 2\}$, and the domain of $g$ is $\{x \mid x \geq 0\}$. The intersection is $[0, 2) \cup (2, \infty)$.

Combined Functions:

• $(f+g)(x) = \frac{1}{x-2} + \sqrt{x}$

• $(f-g)(x) = \frac{1}{x-2} - \sqrt{x}$

• $(fg)(x) = \frac{\sqrt{x}}{x-2}$

• $\bigg(\frac{f}{g}\bigg)(x) = \frac{1}{(x-2)\sqrt{x}}$ (domain excludes $x=0$ and $x=2$)

Composition of Functions

Definition and Notation

The composition of two functions f and g is a new function defined by applying g first, then f to the result:

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

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

Example: Composition

• Let $f(x) = \sqrt{x}$ and $g(x) = x^2 + 1$. Then $h(x) = f(g(x)) = \sqrt{x^2 + 1}$.

Machine Diagram:

First, apply $g$ (square and add 1), then apply $f$ (take the square root). This process can be visualized as a sequence of operations.

Properties of Composition

• Composition is generally not commutative: $f \circ g \neq g \circ f$ in most cases.

• Composition can be extended to three or more functions: $f \circ g \circ h(x) = f(g(h(x)))$.

Example: Composition of Three Functions

• Let $f(x) = x^2 + 1$, $g(x) = x$, $h(x) = x + 3$. Then $f(g(h(x))) = f(g(x+3)) = f(x+3) = (x+3)^2 + 1$.

Decomposing Functions

Recognizing Composition

Sometimes, a complicated function can be expressed as a composition of simpler functions. For example, given $F(x) = \sqrt{x+9}$, we can write:

• $g(x) = x + 9$

• $f(x) = \sqrt{x}$

• $F(x) = f(g(x))$

Applications of Composition

Modeling Real-World Situations

Composition is useful for modeling scenarios where one quantity depends on another, which in turn depends on a third variable. For example, if the radius $R$ of a balloon depends on the volume $V$ of air, and $V$ depends on time $t$, then $R$ is a function of $t$ via composition.

Example: Distance and Time

• A ship travels at 20 mi/h parallel to a shoreline, passing a lighthouse at noon. The distance $s$ from the lighthouse after time $t$ is found by composing two functions:

• $s = f(d) = \sqrt{25 + d^2}$, where $d$ is the distance traveled since noon.

• $d = g(t) = 20t$

• Thus, $s = f(g(t)) = \sqrt{25 + (20t)^2}$

Summary Table: Function Combinations

Operation	Formula	Domain
Sum	$f(x) + g(x)$	$A \bigcap B$
Difference	$f(x) - g(x)$	$A \bigcap B$
Product	$f(x)g(x)$	$A \bigcap B$
Quotient	$\frac{f(x)}{g(x)}$	$\{x \in A \bigcap B \mid g(x) \neq 0\}$
Composition	$f(g(x))$	$\{x \mid x \in \text{Dom}(g), g(x) \in \text{Dom}(f)\}$

Additional info: The notes include both definitions and practical examples, as well as a real-world application involving distance and time. The machine diagram and arrow diagram are referenced to help visualize composition, but are not reproduced here.