Relations, Functions, Domain, and Range: College Algebra Study Notes

Relations and Functions

Definitions and Key Concepts

Understanding the difference between relations and functions is foundational in College Algebra. These concepts describe how sets of numbers (inputs and outputs) are paired together.

• Relation: A relation is a connection between x and y values. Graphically, relations are represented as ordered pairs $(x, y)$.

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

Example: The set $ig\{(1,2), (2,4), (3,6)\big\}$ is a function because each x-value is paired with only one y-value. The set $ig\{(1,2), (1,3), (2,4)\big\}$ is not a function because 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 at more than one point, the graph does not represent a function.

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

Example: A parabola opening upwards passes the vertical line test, but a circle does not.

Inputs and Outputs

Mapping Diagrams

Mapping diagrams visually show how each input is paired with an output.

• Each input should map to only one output for the relation to be a function.

Example:

Inputs (x)	Outputs (y)
-2	2
1	4
3	-2

This mapping is a function because each input has only one output.

Verifying if Equations are Functions

Algebraic Approach

To verify if an equation is a function, solve for y and check if each x-value produces only one y-value.

• Given a graph: Use the vertical line test.

• Given an equation: Check if any x-value results in multiple y-values.

Example:

Equation	Function?
$y = 4 + 3x$	Function (each x gives one y)
$x^2 + y^2 = 25$	Not a function (some x give two y values)

Equations with even powers of y (like $y^2$) often are not functions.

Function Notation

Writing Functions

If an equation is a function, it can be written in function notation by replacing y with f(x).

• Example: $y = -2x + 10$ becomes $f(x) = -2x + 10$

• To evaluate, substitute the value for x: $f(3) = -2(3) + 10 = 4$

Domain and Range

Definitions

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

• To find the domain, look at the extent of the graph along the x-axis.

• To find the range, look at the extent of the graph along the y-axis.

Interval and Set Builder Notation

• Interval Notation: Uses brackets and parentheses to show which values are included or excluded. Example: $[1,5)$ means x is at least 1 and less than 5.

• Set Builder Notation: Uses inequalities. Example: $\{x | 1 \leq x < 5\}$

Example: For a graph with domain $[-4,0) \cup [1,4]$ and range $(-3,1] \cup [1,3]$.

Finding the Domain of an Equation

Restrictions

Some equations have restrictions that limit the domain:

• Square Roots: The expression inside the square root must be non-negative.

• Denominators: The denominator cannot be zero.

1) Inside a Square Root

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

Example: $f(x) = \sqrt{x+4}$, restriction: $x+4 \geq 0 \Rightarrow x \geq -4$

Domain: $[-4, \infty)$

2) x in the Denominator of a Fraction

For $f(x) = \frac{1}{x-5}$, the domain is all real numbers except $x = 5$.

Domain: $(-\infty, 5) \cup (5, \infty)$

Summary Table: Function vs. Not a Function

Type	Definition	Test	Example
Function	Each input has one output	Passes vertical line test	$y = 2x + 1$
Not a Function	Some inputs have multiple outputs	Fails vertical line test	$x^2 + y^2 = 25$

Practice Problems

• Determine if a relation is a function by mapping inputs to outputs.

• Use the vertical line test on graphs to identify functions.

• Find the domain and range of a graph using interval notation.

• Find the domain of equations with square roots and denominators.

Additional info: These notes cover foundational concepts from College Algebra Chapter 1: Graphs, Functions, and Models, including definitions, graphical tests, and domain/range analysis.