2.1 Represent Relations and Functions

 

Relation

A mapping, or pairing, of input values with output values

 

Domain

The set of input values in a relation

 

Range

The set of output values in a relation

 

Function

A relation for which each input has exactly one output

 

Equation in two variables

An equation that has an independent or input variable and a dependent or output variable that depends on the value of the input variable

 

Linear function

A function that can be written in the form y = mx+ b where m and b are constants

 

REPRESENTING RELATIONS

 

A relation can be represented in the following ways:

Ordered Pairs

Table

Graph

Mapping Diagram

(-2, 2)

(-2, -2)

(0, 1)

(3, 1)

x

y

-2

2

-2

-2

0

1

3

1

Rounded Rectangle: -2 2 1 Rounded Rectangle: -2 0 3 Input       Output

 

 

Example 1

Identify functions

 

Tell whether each relation is a function. Explain.

a. Input         Output             b. Input           Output

 


Solution

a.   The relation __is__ a function because each input is mapped onto _exactly one__ output.

b.   The relation _is not_ a function because the input __2___ is mapped onto _2_ and _3_.

 

Complete the following exercise.

 

1.      Is the relation given by the ordered pairs (-5, 2), (-3, -1), (0, 0), (0, 2) and (0, 5) a function? Explain.

No, the relation is not a function because the value 0 maps to 0, 2 and 5.

 

Vertical Line Test

 

A relation is a function if and only if no _vertical_ line intersects the graph of the relation at more than _one point_.

Function                                                             Not a function

 

 

 

Example 2

Use the vertical line test

 

Is the relation represented by the graph a function? Explain.

a.                                           b.

 

 

Solution

a.This graph _does_ represent a function because no vertical line intersects the graph at more than _one point__.

b.   This graph _does not_ represent a function because the vertical lines at x = _3_ and at x = _6_ intersect the graph at more than one point.

 

GRAPHING EQUATIONS IN TWO VARIABLES

 

To graph an equation in two variables, follow these steps:

Step 1 Construct a table of _values_.

 

Step 2 Plot enough points from the table to recognize a _pattern_.

 

Step 3 Connect the points with a __line__ or _curve_.

 

Example 3

Graph an equation in two variables

 

Graph the equation y = -2x - 2.

Solution

 

 

 

 


Step 1 Construct a table of values.

x

-2

-1

0

1

2

y

__2__

__0__

__-2_

_-4__

__-6_

 

Step 2 Plot the points. Notice that they all lie on a _line_.

 

Step 3 _Connect_ the points with a line.

 

Example 4

Classify and evaluate functions

 

Tell whether the function is linear. Then evaluate the function when x = -3.

a. f(x) = 6x + 10                          b. g(x) = 2x2 + 4x -1

 

Solution

a.      The function f is _linear_ because it has the form f(x) = mx + b.

f(x) = 6x + 10                                     Write function.

f(__-3__) = 6(_-3__) + 10                Substitute __-3__ for x.

= __-8__                           Simplify.

b.       The function g is _not linear_ because it has an x2-term.

g(x) = 2x2 + 4x - 1                                               Write function.

g(_-3__) = 2(__-3__)2 + 4(__-3__) -1               Substitute __-3__ for x.

= __5__                                             Simplify.

 

Complete the following exercises.

2.    Use the vertical line test to tell whether the relation is a function.

 


is a function

 

3.    Graph the equation y = 2x - 3.