Relations, Functions, Domain, and Range in College Algebra

Relations and Functions

Definitions and Key Concepts

Understanding the difference between relations and functions is fundamental in College Algebra. These concepts help describe how variables interact and are represented graphically and algebraically.

• Relation: A relation is a connection between two sets of values, typically called inputs and outputs. Graphically, relations are represented as pairs $(x, y)$.

• Function: A function is a special type of relation where each input has at most one output. This means for every $x$ value, there is only one corresponding $y$ value.

• Example: The set $ig\{(1,2), (2,3), (3,4)\big\}$ is a function, but $ig\{(1,2), (1,3), (2,4)\big\}$ is not, since $x=1$ maps to two different $y$ values.

Function Notation

Functions are often written using function notation, such as $f(x)$, which denotes the output when $x$ is the input.

• Example: If $f(x) = 2x + 1$, then $f(3) = 2(3) + 1 = 7$.

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 graph does not represent a function.

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

• Example: The graph of $y = x^2$ passes the vertical line test, but the graph of a circle $x^2 + y^2 = r^2$ does not.

Inputs and Outputs

Identifying Inputs and Outputs

In a relation or function, the input is typically the $x$ value, and the output is the $y$ value.

• Example: For the relation $\{(-3,5), (0,2), (3,5)\}$, inputs are $-3, 0, 3$ and outputs are $5, 2, 5$.

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

Verifying if Equations are Functions

Graphical and Algebraic Methods

To verify if an equation is a function, use either the graph or the equation itself:

• Graph: Apply the vertical line test.

• Equation: Check if any $x$ value results in multiple $y$ values.

Decision Table

Method	Test	Result
Graph	Fails Vertical Line Test	Not a Function
Graph	Passes Vertical Line Test	Function
Equation	Any $x$ gives multiple $y$	Not a Function
Equation	Each $x$ gives one $y$	Function

• Example: $y + 4 = 3x$ is a function because for each $x$, there is one $y$.

• Example: $x^2 + y^2 = 25$ is not a function because for some $x$, there are two $y$ values.

Domain and Range

Definitions

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

• Domain: Allowed $x$ values.

• Range: Allowed $y$ values.

Finding Domain and Range from a Graph

To find the domain, "squish" the graph to the $x$-axis; to find the range, "squish" to the $y$-axis.

• Interval Notation: $(a, b)$ means all values between $a$ and $b$, not including endpoints. $[a, b]$ includes endpoints.

• Set Builder Notation: $\{x | a < x < b\}$ means $x$ is between $a$ and $b$.

• Example: If a graph extends from $x = -4$ to $x = 5$, domain is $[-4, 5]$.

• If a graph extends from $y = -1$ to $y = 2$, range is $[-1, 2]$.

Notation Table

Type	Example	Includes Endpoints?
Interval Notation	$[a, b]$	Yes
Interval Notation	$(a, b)$	No
Set Builder Notation	$\{x | a < x < b\}$	No
Set Builder Notation	$\{x | a \leq x \leq b\}$	Yes

Union Symbol

When there are multiple intervals, use the union symbol $\cup$ to combine them.

• Example: $(-\infty, -2) \cup (2, \infty)$

Finding the Domain of an Equation

Restrictions

Some equations have restrictions on the domain due to square roots or denominators.

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

• Fraction: The denominator cannot be zero.

Examples

• Square Root Example: For $f(x) = \sqrt{x}$, domain is $[0, \infty)$.

• Fraction Example: For $f(x) = \frac{2}{x-5}$, domain is $(-\infty, 5) \cup (5, \infty)$.

Practice Problems

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

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

Summary Table: Function vs. Not a Function

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

Additional info: This guide expands on the original notes by providing full definitions, examples, and tables for clarity and completeness, suitable for College Algebra students preparing for exams.