Domain and Range in Functions: College Algebra Study Notes

Domain and Range of Functions

Definitions and Interval Notation

Understanding domain and range is fundamental in analyzing functions in College Algebra. The domain of a function consists of all possible input values (x-values), while the range consists of all possible output values (y-values). Both are typically expressed in interval notation.

• Domain: The set of all x-values included in the graph of a function. Expressed in interval notation, e.g., $(-\infty, \\infty)$.

• Range: The set of all y-values included in the graph of a function. Also expressed in interval notation, e.g., $(-\\infty, \\infty)$.

How to Determine Domain and Range from a Graph

To find the domain and range from a graph, follow these steps:

1. Label all arrows: Indicate directions (Up, Down, Right, Left).

2. For domain: Examine the graph from left to right, noting the lowest and highest x-values included.

3. For range: Examine the graph from bottom to top, noting the lowest and highest y-values included.

4. Infinity: Use $-\\infty$ for negative infinity and $+\\infty$ for positive infinity in interval notation.

5. Brackets and Parentheses: Use [ ] for values included in the graph, ( ) for values not included (open intervals).

Special Cases and Graph Features

• Arrows: Indicate the graph continues indefinitely in a direction (use infinity in interval notation).

• Holes: Points not included in the graph (exclude these values from the interval).

• Vertical Lines: Domain is a single x-value; range is all y-values.

• Horizontal Lines: Range is a single y-value; domain is all x-values.

Examples and Applications

Below are examples illustrating how to determine domain and range from various graphs:

Example 1: Continuous Graph with No Holes

• Domain: $(-\\infty, \\infty)$ (graph extends left and right without breaks)

• Range: $(-\\infty, \\infty)$ (graph extends up and down without breaks)

Example 2: Graph with a Hole

• Domain: $(-\\infty, 2) \\cup (2, \\infty)$ (hole at $x=2$)

• Range: $(-\\infty, 1) \\cup (1, \\infty)$ (hole at $y=1$)

Example 3: Vertical Line

• Domain: $\{5\}$ (only $x=5$ is included)

• Range: $(-\\infty, \\infty)$ (all y-values)

Example 4: Horizontal Line

• Domain: $(-\\infty, \\infty)$ (all x-values)

• Range: $\{2\}$ (only $y=2$ is included)

Table: Domain and Range Scenarios

Graph Type	Domain	Range
Continuous, no holes	$(-infty, infty)$	$(-infty, infty)$
Hole at $x=a$	$(-infty, a) cu (a, infty)$	Depends on y-values at the hole
Vertical line $x=a$	$\{a\}$	$(-infty, infty)$
Horizontal line $y=b$	$(-infty, infty)$	$\{b\}$

Additional info:

• When a graph has multiple holes, exclude each corresponding value from the domain or range using union notation.

• Always check for arrows, holes, and labeled points to determine the correct intervals.