Introduction to Functions Understand the definition of a relation. Understand the definition of a function. Decide whether an equation defines a function

Introduction to Functions

Understand the definition of a relation.

Understand the definition of a function.

Decide whether an equation defines a function.

Find domains and ranges.

Use function notation. Apply the function concept in an application.

3.6

2

3

4

5

6

1

Understand the definition of a relation.

In an ordered pair (x, y), x and y are called the components of the ordered pair.

Any set of ordered pairs is called a relation.

The set of all first components of the ordered pairs of a relation is the domain of the relation, and the set of all second components of the ordered pairs is the range of the relation.

Solution:

Domain:

Range:

Identify the domain and range of the relation.

4,0 , 6,1 , 7,1 , 3,2

EXAMPLE 1 Identifying Domains and Ranges of Relations

0,1,2

3,4,6,7

A very important type of relation called a function.

By definition, the relation in the following order pairs is not a function, because the same first component, 3, corresponds to more then one second component.

5 63 3 3, , , ,7 3,8

If the ordered pairs from this example were interchanged, giving the relation

the result would be a function.

In that case, each domain element (first component) corresponds to exactly one range element (second component).

5 6 7, , , ,3 3 3 8,3 ,

Understand the definition of a function.

FunctionA function is a set of ordered pairs in which each first component corresponds to exactly one second component.

Mapping

Relations and functions can also be expressed as a correspondence or mapping from one set to another. In the example below the arrows from 1 to 2 indicates that the ordered pair (1, 2) belongs to F. Each first component is paired with exactly one second component.

1– 23

24

– 1

x-axis values y-axis values

Mapping

In the mapping for relations H, which is not a function, the first component – 2 is paired with two different second components, 1 and 0.

– 4– 2

10

x-axis values y-axis values

Solution: function

Determine whether each relation is a function.

2,8 , 1,1 , 0,0 , 1,1 , 2,8 ,

5,2 , 5,1 , 5,0

Solution: not a function

EXAMPLE 2 Determining Whether Relations Are Functions

Decide whether an equation defines a function.

Given the graph of an equation, the definition of a function can be used to decide whether or not the graph represents a function. By the definition of a function, each x-value must lead to exactly one y-value.

Vertical Line TestIf a vertical line intersects a graph in more than one point, then the graph is not the graph of a function.

Any nonvertical line is the graph of a function. For this reason, any linear equation of the form y = mx + b defines a function. (Recall that a vertical line has an undefined slope.)

USING THE VERTICAL LINE TEST

a.

(– 1, 1)

(1, 2)

(0, – 1)

(4, – 3)

This graph represents a function.

Use the vertical line test to determine whether each relation graphed is a function.

x

y

Determine whether each relation is a function.

Solution: functionSolution: not a function

EXAMPLE 3 Determining Whether Relations Define Functions

b.

This graph fails the vertical line test, since the same x-value corresponds to two different y-values; therefore, it is not the graph of a function.

4– 4

6

– 6

x

y



c.


x

y



d.


x

y



By the definitions of domain and range given for relations, the set of

all numbers that can be used as replacements for x in a function is the

domain of the function. The set of all possible values of y is the range

of the function.

Find domains and ranges.

FINDING DOMAINS AND RANGES OF RELATIONS

Give the domain and range of the relation. Tell whether the relation defines a function.

a. (3, 1),(4,2),(4,5),(6,8)

The domain, the set of x-values, is {3, 4, 6}; the range, the set of y-values is {– 1, 2, 5, 8}. This relation is not a function because the same x-value, 4, is paired with two different y-values, 2 and 5.



b. 467

– 3

100200300

The domain is {4, 6, 7, – 3}; the range is {100, 200, 300}. This mapping defines a function. Each x-value corresponds to exactly one y-value.



c. This relation is a set of ordered pairs, so the domain is the set of x-values {– 5, 0, 5} and the range is the set of y-values {2}. The table defines a function because each different x-value corresponds to exactly one y-value.

x y

– 5 2

0 2

5 2

FINDING DOMAINS AND RANGES FROM GRAPHS

Give the domain and range of each relation.

a.

(– 1, 1)

(1, 2)

(0, – 1)

(4, – 3)

The domain is the set of x-values which are {– 1, 0, 1, 4}. The range is the set of y-values which are {– 3, – 1, 1, 2}.

x

y



b.The x-values of the points on the graph include all numbers between – 4 and 4, inclusive. The y-values include all numbers between – 6 and 6, inclusive. 4– 4

6

– 6

x

y

The domain is [– 4, 4]. The range is [– 6, 6].



c.The arrowheads indicate that the line extends indefinitely left and right, as well as up and down. Therefore, both the domain and the range include all real numbers, written (– , ).

x

y



d. The arrowheads indicate that the line extends indefinitely left and right, as well as upward. The domain is (– , ). Because there is at least y-value, – 3, the range includes all numbers greater than, or equal to – 3 or [– 3, ).

x

y

Find the domain and range of the function y = x2 + 4.

Solution:

Domain:

,

Range:

4,

EXAMPLE 4 Finding the Domain and Range of Functions

The letters f, g, and h are commonly used to name functions. For example, the function y = 3x + 5 may be written

where f (x), which represents the value of f at x, is read “f of x.” The notation f (x) is another way of writing y in a function. For the function defined by f (x) = 3x + 5, if x = 7, then

Read this result, f (7) = 26, as “f of 7 equals 26.” The notation f (7) means the values of y when x is 7. The statement f (7) = 26 says that the value of y = 26 when x is 7. It also indicates that the point (7,26) lies on the graph of f.

3 5,f x x

7 73 5f

21 5 26.

The notation f (x) does not mean f times x; f (x) means the value of x for the function f. It represents the y –value that corresponds to x in the function f.

Use function notation.

Function NotationIn the notation f (x),

f is the name of the function,x is the domain value,

and f (x) is the range value y for the domain value x.

Use function notation. (cont’d)

Solution:

Find f (−1), for the function.

f (x) = 6x − 2

6 21 1f

6 21f

1 8f

EXAMPLE 5 Using Function Notation

USING FUNCTION NOTATION

For each function, find (3).

Solution For = {( – 3, 5), (0, 3), (3, 1), (6, – 9)}, we want (3), the y-value of the ordered pair where x = 3. As indicated by the ordered pair (3, 1), when x = 3, y = 1,so(3) = 1.

b. ( 3,5),(0,3),(3,1),(6, 1)

Increasing, Decreasing, and Constant Functions

Suppose that a function is defined over an interval I. If x1 and x2 are in I, a increases on I if, whenever x1 < x2, (x1) < (x2)b decreases on I if, whenever x1 < x2, (x1) > (x2)c is constant on I if, for every x1 and x2, (x1) = (x2)

DETERMINING INTERVALS OVER WHICH A FUNCTION IS INCREASING, DECREASING, OR CONSTANT

Determine the intervals over which the function is increasing, decreasing, or constant.

6

2

1 3– 2x

y

DETERMINING INTERVALS OVER WHICH A FUNCTION IS INCREASING, DECREASING, OR CONSTANT

Determine the intervals over which the function is increasing, decreasing, or constant.

6

2

1 3– 2

On the interval (– , 1), the y-values are decreasing; on the interval [1,3], the y-values are increasing; on the interval [3, ), the y-values are constant (and equal to 6).

Solution

x

y

The median age at first marriage for women in the United States for selected years is given in the table.

7 73 5f Write a set of ordered pairs that defines a function f for these data.

Give the domain and range of f.

Find f (2006).

EXAMPLE 6 Applying the Function Concept to Population

Solution:

25.9

range: 25.1, 25.3, 25,9 domain: 2000, 2002, 2004, 2006

2000,25.1 2002,25.3 2004,25.3 2006,25.9f

http://www.khanacademy.org/math/algebra/algebra-functions/v/introduction-to-functions

http://www.khanacademy.org/math/algebra/algebra-functions/v/testing-if-a-relationship-is-a-function

http://www.khanacademy.org/math/algebra/algebra-functions/v/functional-relationships-1

Khan Academy Videos

http://www.khanacademy.org/math/algebra/algebra-functions/v/graphing-a--basic-function

http://www.khanacademy.org/math/algebra/algebra-functions/v/domain-and-range-of-a-function

http://www.khanacademy.org/math/algebra/algebra-functions/v/domain-and-range-1

http://www.khanacademy.org/math/algebra/algebra-functions/v/domain-and-range-2



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Introduction to Functions Understand the definition of a relation. Understand the definition of a function. Decide whether an equation defines a function