Relations and Functions

A relation is a rule that takes an input from a set (called the domain) and gives one or more outputs of another set (called the range).

Examples

1.         (0,0), (4,4), (0,3), (2,1)

2.         y = 2x

3. The circle is a relation.  We can go from an x-coordinate to the y-coordinates of the circle that have that x-coordinate.

A function is a relation such that for every input there is exactly one output.

Examples:

Examples 2 and 3 above are functions, whereas examples 1 and 4 are not functions.

Determining the Domain of a Function

Rule 1:  The domain of a polynomial is the set of all real numbers.

Example:

For

f(x)  =  3x2 + 2x - 1

the domain is the set of all real numbers.

Rule 2:  The domain of a rational function (poly)/(poly) is the set of all real numbers except where the denominator is 0.

Example:

For

x - 1
f(x)   =
x + 1

the domain is all real numbers except where x  =  -1.

Rule 3:  The domain of a square root function is all real numbers that make the inside of the square root positive.

Example:

Find the domain of

Solution:

We set up  the inequality

x2 - x - 6  >  0

and use our steps of quadratic inequalities to solve. Factoring we get

(x - 3) (x + 2)  >  0

Putting -2 and 3 on a number line, gives three regions.  The table shows

 x+2 x - 3 Total Left (-3) - - + Middle(0) + - - Right(4) + + +

Hence the solution is

(-,-2]     U     [3,)

The Vertical Line Test

If every vertical line passes through the graph at most once then the graph is the graph of a function.

Example

The Circle is not the graph of a function since there are vertical lines that cross it twice.