BackFunctions and Graphs: Definitions, Examples, and Domain/Range
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Functions and Graphs
Definition of a Function
A function is a correspondence between a first set, called the domain, and a second set, called the range, such that each element of the domain is paired with exactly one element of the range.
Domain: The set of all possible input values (independent variable).
Range: The set of all possible output values (dependent variable).
It is important to note that not every correspondence between two sets is a function.
Examples: Determining Functions
Example: Given a set of U.S. Presidents and Supreme Court Justices, if each President is paired with exactly one Justice, the correspondence is a function. If a President is paired with more than one Justice, it is not a function.
Non-Function Example: If an input (from the domain) is paired with more than one output (from the range), the correspondence is not a function.
Definition of a Relation
A relation is a correspondence between a first set, called the domain, and a second set, called the range, that pairs each element of the domain with at least one element of the range. Every function is a relation, but not every relation is a function.
Identifying Functions from Ordered Pairs
Given a set of ordered pairs, the relation is a function if no input value (first element) is repeated with a different output value (second element).
Example: The set { (1, -1), (2, 3), (3, 4) } is a function because each input is paired with only one output.
Non-Example: The set { (1, 2), (1, 3) } is not a function because the input 1 is paired with two different outputs.
Notation for Functions
Functions are often written as equations, such as . The input variable is usually , and the output is or . The notation denotes the value of the function at .
Example: If , then .
The Vertical Line Test
The vertical-line test is a graphical method to determine if a graph represents a function. If any vertical line intersects the graph at more than one point, the graph does not represent a function.
Example: The graph of a circle fails the vertical-line test and is not a function.
Example: The graph of passes the vertical-line test and is a function.
Finding Function Values
To find , substitute into the function for .
Example: If , then .
Domain and Range of Functions
The domain of a function is the set of all input values for which the function is defined. The range is the set of all possible output values.
For rational functions, exclude values that make the denominator zero.
For square root functions, the expression under the root must be non-negative.
Examples:
For , the domain is all real numbers except .
For , the domain is .
Sample Table: Function vs. Relation
Input (Domain) | Output (Range) | Is it a Function? |
|---|---|---|
1 | 2 | Yes |
1 | 3 | No (input 1 has two outputs) |
2 | 4 | Yes |
Summary
A function pairs each input with exactly one output.
The vertical-line test helps determine if a graph is a function.
Domain and range are fundamental concepts for understanding functions.
Function notation and evaluation are essential skills in algebra.
