Introduction to Functions and Their Graphs: Relations, Functions, Domain, and Range

Intro to Functions & Their Graphs

Relations and Functions

A foundational concept in precalculus is understanding the difference between relations and functions, and how to represent them graphically and algebraically.

• Relation: A connection between a set of x (input) and y (output) values. Graphically, relations are represented as ordered pairs (x, y).

• Function: A special type of relation where each input has exactly one output.

Examples:

• If the set of ordered pairs is {(1, 2), (2, 3), (3, 4)}, each input (x-value) is paired with only one output (y-value), so this is a function.

• If the set is {(1, 2), (1, 3), (2, 4)}, the input 1 is paired with two different outputs (2 and 3), so this is not a function.

Visual Representation: Functions and relations can be shown using mapping diagrams (inputs and outputs) or graphs on the coordinate plane.

Vertical Line Test

The Vertical Line Test is a graphical method to determine if a graph represents a function:

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

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

Example: The graph of a parabola (y = x2) passes the vertical line test and is a function. A circle does not pass the test and is not a function.

Verifying if Equations are Functions

To verify if an equation is a function, solve for y in terms of x:

• If each input x gives only one output y, it is a function.

• If any input x gives more than one output y, it is not a function.

Example:

• y = x + 3 is a function because each x gives one y.

• x = y2 is not a function because each x gives two possible y values (positive and negative square roots).

Domain and Range

Finding the Domain and Range of a Graph

The domain of a graph 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, look for the leftmost and rightmost points on the graph.

• To find the range, look for the lowest and highest points on the graph.

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 | -2 ≤ x < 5}.

Example: For a graph that starts at x = -2 and ends at x = 4, the domain is [-2, 4]. If the lowest y-value is 0 and the highest is 5, the range is [0, 5].

When there are multiple intervals or jumps in the graph, use the union symbol ( ∪ ).

Finding the Domain of an Equation

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

• Inside a Square Root: The expression inside the square root must be greater than or equal to zero.

• In the Denominator of a Fraction: The denominator cannot be zero.

Examples:

• For $f(x) = \sqrt{x-2}$, set $x-2 \geq 0 \implies x \geq 2$. So, the domain is [2, ∞).

• For $f(x) = \frac{1}{x-3}$, set $x-3 \neq 0 \implies x \neq 3$. So, the domain is $(-\infty, 3) \cup (3, \infty)$.

Practice Problems

• Given a set of ordered pairs, identify the inputs and outputs, and determine if it is a function.

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

• Given an equation, solve for y and determine if it is a function.

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

• Find the domain of a function given by an equation, especially when involving square roots or denominators.

Summary Table: Function vs. Not a Function

Type	Definition	Example
Function	Each input has exactly one output	y = x + 2
Not a Function	At least one input has more than one output	x = y2

Additional info: These notes cover the foundational concepts of functions, relations, domain, and range, which are essential for further study in precalculus and calculus.