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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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