Relations, Functions, and Domain & Range in College Algebra

Relations and Functions

Definitions and Key Concepts

Understanding the distinction between relations and functions is fundamental in College Algebra. This section introduces the basic definitions and graphical representations.

• Relation: A relation is a connection between two sets of values, typically represented as ordered pairs (x, y).

• Function: A function is a special type of relation in which each input (x-value) is associated with at most one output (y-value).

• Graphical Representation: Relations and functions can be visualized on the coordinate plane, where each point corresponds to an (x, y) pair.

Example: The set { (1,2), (2,3), (3,4) } is a function, but { (1,2), (1,3) } is not, since the input 1 is paired with two different outputs.

Vertical Line Test

The Vertical Line Test is a graphical method to determine if a relation is a function.

• If any vertical line crosses the graph more than once, the relation is not a function.

• If every vertical line crosses the graph at most once, the relation is a function.

Example: The graph of a parabola (y = x2) passes the vertical line test, but a circle (x2 + y2 = r2) does not.

Identifying Functions from Sets and Graphs

Inputs and Outputs

For a given set of ordered pairs, the inputs are the x-values and the outputs are the y-values.

• To determine if a set is a function, check if any input is paired with more than one output.

Example: For the set { (2,5), (0,2), (2,9) }, the input 2 is paired with both 5 and 9, so this is not a function.

Function or Not? (Graphical)

Given a graph, use the vertical line test to determine if it represents a function.

• Graphs such as lines and parabolas are functions.

• Graphs such as circles are not functions.

Verifying if Equations are Functions

Algebraic and Graphical Methods

To verify if an equation defines y as a function of x, use the following approaches:

• Graphical: Does the graph fail the vertical line test?

• Algebraic: Does any x-value result in multiple y-values?

Example:

x	y (A: y + 4 = 3x)	y (B: x2 + y2 = 25)
-1	-7	y = ±√(25 - 1) = ±√24
0	4	y = ±5
1	7	y = ±√(25 - 1) = ±√24
2	10	y = ±√(25 - 4) = ±√21

• If the equation has y to the first power and can be solved uniquely for y, it is a function.

• If the equation has y squared (or higher even powers), it may not be a function.

Function Notation and Evaluation

Writing and Evaluating Functions

Functions are often written in the form f(x). To evaluate, substitute the given value for x.

• Example: If f(x) = -2x + 10, then f(3) = -2(3) + 10 = 4.

Domain and Range

Definitions

The domain of a function is the set of all possible input (x) values, and the range is the set of all possible output (y) values.

• To find the domain, "squish" the graph onto the x-axis.

• To find the range, "squish" the graph onto the y-axis.

Interval and Set Builder Notation

• Interval Notation: Uses parentheses ( ) for values not included, and brackets [ ] for values included.

• Set Builder Notation: Describes the set using inequalities, e.g., { x | -4 ≤ x ≤ 5 }.

Example: If a graph extends from x = -4 to x = 5, the domain is [-4, 5]. If the y-values range from -1 to 2, the range is [-1, 2].

Union Symbol

• When a graph has multiple intervals, use the union symbol ( ∪ ) to combine them in interval notation.

Finding the Domain of an Equation

Restrictions on the Domain

When given an equation, determine the domain by identifying values that make the function undefined. Two common cases:

1. Inside a Square Root: The expression inside the square root must be non-negative (≥ 0).

2. In the Denominator: The denominator cannot be zero.

Example 1: Square Root

• For $f(x) = \sqrt{x}$, the domain is $x \geq 0$.

• Expressed in interval notation: [0, ∞)

Example 2: Denominator

• For $f(x) = \frac{2}{x-5}$, the domain is all real numbers except $x = 5$.

• Expressed in interval notation: $(-\infty, 5) \cup (5, \infty)$

Practice Problems

• Find the domain of $f(x) = \sqrt{x^2 + 4}$.

• Find the domain of $f(x) = \frac{1}{x^2 - 5x + 6}$.

Summary Table: Function Tests and Domain Restrictions

Situation	Test/Restriction	Result
Graph	Vertical Line Test	Pass: Function; Fail: Not a Function
Equation (y to first power)	Unique y for each x	Function
Equation (y squared or higher even power)	Multiple y for some x	Not a Function
Square Root	Expression inside ≥ 0	Domain: Set where expression is non-negative
Denominator	Denominator ≠ 0	Domain: All real numbers except where denominator is zero

Additional info: These notes cover foundational concepts in College Algebra, including relations, functions, domain, range, and function notation, with both graphical and algebraic perspectives. Practice problems and examples are included to reinforce understanding.