BackRelations and Functions: Domain, Range, and Function Notation
Study Guide - Smart Notes
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Relations and Functions
Definition of a Relation
A relation in mathematics is a set of ordered pairs (x, y). Each ordered pair consists of an input value (x) and an output value (y). Relations can be represented as sets of points on a coordinate plane.
Ordered Pair: A pair of numbers written in the form (x, y), where x is the input (independent variable) and y is the output (dependent variable).
Example: The set { (1, 2), (2, 3), (3, 4) } is a relation.
Domain and Range
The domain of a relation is the set of all possible input values (x-values). The range is the set of all possible output values (y-values).
Domain: All x-values from the set of ordered pairs.
Range: All y-values from the set of ordered pairs.
Example: For the relation { (1, 2), (2, 3), (3, 4) }, the domain is {1, 2, 3} and the range is {2, 3, 4}.
Function Notation and Determining Functions
A function is a special type of relation in which each input value (x) is paired with exactly one output value (y). To determine if a relation is a function, check that no x-value is repeated with a different y-value.
Function: A relation where each element of the domain is paired with exactly one element of the range.
Vertical Line Test: If any vertical line crosses the graph of the relation more than once, the relation is not a function.
Example: The relation { (1, 2), (2, 3), (3, 4) } is a function, but { (1, 2), (1, 3), (2, 4) } is not, because x = 1 is paired with two different y-values.
Summary Table: Relation vs. Function
Term | Definition | Example |
|---|---|---|
Relation | Any set of ordered pairs (x, y) | { (1, 2), (2, 3), (3, 4) } |
Function | Each x-value is paired with only one y-value | { (1, 2), (2, 3), (3, 4) } |
Not a Function | At least one x-value is paired with more than one y-value | { (1, 2), (1, 3), (2, 4) } |
Key Formulas
Domain:
Range:
Example Problem
Given: A set of points plotted on a coordinate grid.
Tasks:
Write the set of ordered pairs (x, y) that defines the relation.
State the domain of the relation.
State the range of the relation.
Determine if the relation defines y as a function of x.
Solution Steps:
List all the points as ordered pairs.
Extract all x-values for the domain.
Extract all y-values for the range.
Check if any x-value is repeated with a different y-value to determine if it is a function.
Additional info: The original image contains a coordinate grid with plotted points, but the specific coordinates are not visible. The above steps and explanations provide a general method for analyzing such relations in College Algebra.
