FUNCTIONS

OBJECTIVES

Define a relation Define a function Define the domain and range of a function Evaluate a function Graph a function

SET OF ORDERED PAIRS

The table shows the number of medals won by United States athletes during five Winter Olympics.

The following is a set of ordered pairs:{(x, y) | x = year, y = number of medals}

{(1992, 11), (1994, 13), (1998, 13), (2002, 34), (2006, 25), (2010, 37)}

DOMAIN AND RANGE OF A RELATION

Relation: set of ordered pairs {(1992, 11), (1994, 13), (1998, 13), (2002, 34), (2006, 25), (2010, 37)}

Domain: set of all first components of a relation{1992, 1994, 1998, 2002, 2006, 2010}

Range: set of all second components of a relation{11, 13, 34, 25, 37}

EXAMPLE

Given the relation {(–2, –5), (4, 7), (8, 9)} Find its domain

{–2, 4, 8} Find its range

{–5, 7, 9}

FUNCTION

FunctionA function is a set of ordered pairs (a relation) in which to each first component, there corresponds exactly one second component.

Is this a function?{(1, 2), (2, 4), (3, 6), (4, 8), (, 10)}

Yes{(2, 4), (3, 9), (4, 16), (5, 25), (6, 36)}

Yes{(36, 6), (25, 5), (25, -5), (16, 4), (9, 3)}

No{(1, 6), (2, 5), (3, -5), (4, 4), (4, 3)}

No

Y AS FUNCTION OX X Given the equation: y = 3x – 4

Since the equation describes a set of ordered pairs, it describes a relation.

Since, for each value of the domain, there is only one value of the range, the equation also describes a function.

Y AS FUNCTION OF X

y is a Function of xAny equation in x and y where each value of x determines exactly one value of y is called a function. In this case, we say that y is a function of x.

y = 3x – 4

Variable y is a function of x.Variable x is the independent variable.Variable y is the dependent variable.

FUNCTION Comments

Function is always a relation, but relation is not always a function.

E.g., {(2, 1), (2, 3)} is a relation, but not a function Another view of a Function

x (independent y (dependent variable) variable)

2 -10 -1 -7 0 -4 1 -1 2 2

y = 3x - 4

(function)

EXAMPLE

Determine whether the equations define y to be a function of x.a.y = x2 b.x = y2

SOLUTION y = x2

The table shows that for each value of x, there is exactly one value of y. So, the relation is a function.

SOLUTION x = y2

We construct a table of ordered pairs for the equation x = y2. Because y is squared, it will be more convenient to substitute values for y and compute the corresponding values for x.

The table shows that for each value of x, there are more than one value of y. Thus, the equation does not describe a function.

FUNCTION NOTATION

Write y = 2x – 3 as f(x) = 2x – 3. Read this as: “f of x = 2x – 3.”

The notation y = f(x), read “y equals f of x,” means that the variable y depends on the value of x.

EXAMPLES

Let f (x) = 2x – 3. Find the followinga. f (3)

f(3) = 2(3) – 3 = 6 – 3 = 3

b. f (–1) f(-1) = 2(-1) – 3 = -5

f (0) f(0) = 2(0) – 3 = -3

a. the value of x for which f (x) = 5. a. f(x) = 2x - 3

5 = 2x – 38 = 2xx = 4

GRAPH OF A FUNCTION

Given: f (x) = | x | Graph the function Determine the domain and range

DOMAIN AND RANGE

Domain—set of all real numbers D = {x | x is a real number}

Range—set of all positive numbers and 0 R = {y | y ≥ 0}

ANOTHER VIEW OF FUNCTION

0149

-2-10123

x

y

Relation

-2-10123

0149

Function

x

y

DETERMINE WHETHER A GRAPH REPRESENTS A FUNCTION

A vertical line test can be used to determine whether the graph of an equation represents a function.

If any vertical line intersects a graph more than once, the graph cannot represent a function, because to one number x, there would correspond more than one value of y.

The graph in Figure represents a function, because every vertical line that intersects the graph does so exactlyonce.

DETERMINE WHETHER A GRAPH REPRESENTS A FUNCTION

The graph in Figure does not represent a function, because some vertical lines intersect the graph more than once.

DETERMINE WHETHER A GRAPH REPRESENTS A FUNCTION

Determine whether each graph represents a function.

EXCEL GRAPHING

MS Excel Graphing functions.xls