A set of ordered pairs is called a relation.

ex.

(1,2), (3,5), (7,10)...are relations.

The first components in the ordered pairs (x-coordinate) is called the domain, and the second components (y-coordinate or *f*(x) coordinate) is called the range.

A **function** is a relation were each member of the domain is reserved one and only one member of the range.

Functions are named by using single letters.

ex.

*f* = {(x,y) | y = x + 4 }

or

*f* (x) = x + 4

The equations above are representations of functions. The second example is the most commonly used representation; it is called the function notation.

The ordered pair for functions is (x, *f* (x)).

consider the example;

*f* (x) = x + 4 at x = 2.*f* (2) = 2 + 4*f* (2) = 6

the ordered pair is (2,6)

ex.

find the range; given the domain {2,4,6} of *f*(x) = 3x - 1

*f*(2) = 3(2) - 1 = 6 -1 = 5*f*(4) = 3(4) - 1 = 12 - 1 = 11*f*(6) = 3(6) - 1 = 18 - 1 = 17

the range is {5,11,17} and the ordered pairs are (2,5); (4,11); (6,17)

Find the domain and range of:

*f*(x) = 2 / (x-3)

Since x cannot be zero; the domain is D = { x| x ¹ 3} and *f*(x) can have any value; thus the range is R = { *f*(x) | any real number}

Graph of Functions: [ to graph functions, just change y into *f*(x) from graphing equations]

Graphing functions is the same as graphing equations with the exception of naming the domain and the range.

Linear Functions:

ex.*f*(x) = 3x + 2:

the function is in the slope-intercept form (y = mx + b)

m(slope) = 3

y-intercept = 2

Quadratic Functions:[ *f*(x) = ax²+ bx + c ]

The same as graphing quadratic equations:

ex.

graph:*f*(x) = 2x²+ 8x + 9

y = 2x²+ 8x + 9 replace *f*(x) with y

y - 9 = 2x+ 8x complete the square

y - 9 + 4 = 2( x²+ 4x + 4)

y - 1 = 2(x + 2)² parabola with vertex at (-2,1)

y = 2(x + 2)²+ 1 solve for y and replace y with f(x)*f*(x) = 2(x + 2)²+ 1

Other functions:

Rules for graphing other functions:

1. Determine the domain of the graph.

2. Check for symmetry.

if

f(-x) =f(x), then symmetric about the y-axis.

iff(-x) = -f(x), then symmetric about the origin.

( functions rule out the possibility that the graph has an x-axis

symmetry)

3. Find the x-intercept and y-intercept of the graph

Evaluate f(0) to find the y-intercept.

To find the x-intercept, find the value or values that will make f(x) = 0.

4. Plot some points for the graph and reflect, according to the symmetry

test.

ex.

graph:

f(x) = | x |the domain is D = {x | x is any real number}

f(-x) = | -x | = | x | =f(x) symmetric about the y-axis.f(0) = | 0 | = 0f(x) = 0 at x = 0

the intercepts are at the origin.

x f(x)1 1 -2 2 -1 1 2 2

**Vertical Line Test:**

Given a graph, it can be determined if it is a function or a relation.

In order for the graph to be a function, the vertical line must only intersect the graph at one and only one point.

**Horizontal Line Test:**

The horizontal line test implies a one to one function, meaning that there is only one value of x associated with each value of f(x).