Algebra II: Functions

Relations and Functions

Relations

A relation is a set of inputs and outputs, often written as ordered pairs (input, output). We can also represent a relation as a mapping diagram or a graph. For example, the relation can be represented as:

Mapping Diagram of Relation

Lines connect the inputs with their outputs. The relation can also be represented as:
Graph of Relation

Functions

A function is a relation in which each input has only one output.

In the relation , y is a function of x , because for each input x (1, 2, 3, or 0), there is only one output y . x is not a function of y , because the input y = 3 has multiple outputs: x = 1 and x = 2 .

Examples:

\: y is a function of x , x is a function of y .

: y is not a function of x ( x = 3 has multiple outputs), x is a function of y .

: y is a function of x , x is not a function of y ( y = 9 has multiple outputs).

: y is not a function of x ( x = 1 has multiple outputs), x is not a function of y ( y = 2 has multiple outputs).

The Line Test for Mapping Diagrams

To check if a relation is a function, given a mapping diagram of the relation, use the following criterion: If each input has only one line connected to it, then the outputs are a function of the inputs.

Example: In the following mapping diagram, y is a function of x , but x is not a function of y :

Line Test

The Vertical and Horizontal Line Tests for Graphs

To determine whether y is a function of x , given a graph of a relation, use the following criterion: if every vertical line you can draw goes through only 1 point, y is a function of x . If you can draw a vertical line that goes through 2 points, y is not a function of x . This is called the vertical line test.

Example 1: In the following graph, y is a function of x :

Passes Vertical Line Test

Example 2: In the following graph, y is not a function of x :

Fails Vertical Line Test

To determine whether x is a function of y , given a graph of a relation, use the following criterion: If every horizontal line you can draw passes through only 1 point, x is a function of y . If you can draw a horizontal line that passes through 2 points, x is not a function of y . This is called the horizontal line test.

Example 1: In the following graph, x is a function of y :

Passes Horizontal Line Test

Example 2: In the following graph, x is not a function of y :

Fails Horizontal Line Test

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