Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice HallCopyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall

Chapter 3

Graphs and Functions


3.2

Introduction to Functions


Relation, Domain, and Range

A relation is a set of ordered pairs.

The domain of the relation is the set of all first components of the ordered pairs.

The range of the relation is the set of all second components of the ordered pairs.


Determine the domain and range of the relation

{(4,9), (–4,9), (2,3), (10, –5)}

Solution

The domain is the set of all the first coordinates of the ordered pairs.

Domain: 4, –4, 2, 10}.

The range is the set of all second coordinates of the ordered pairs.

Range: 9, 3, –5}.

Example 1a


Input (Animal)

Polar Bear

Cow

Chimpanzee

Giraffe

Gorilla

Kangaroo

Red Fox

Output (Life Span)

20

15

10

7

Find the domain and range of the following relation.

Example 1b

Domain: {Polar Bear, Cow, Chimpanzee, Giraffe, Gorilla, Kangaroo, Red Fox}.

Range: {20, 15, 10, 7}.


Some relations are also functions.

A function is a relation in which each first component in the ordered corresponds to exactly one second component.


Given the relation {(4,9), (–4,9), (2,3), (10, –5)}, is it a function?

Solution

Since each element of the domain is paired with only one element of the range, it is a function.

Note: It is okay for a y-value to be assigned to more than one x-value, but an x-value cannot be assigned to more than one y-value (it has to be assigned to ONLY one y-value).

Example 2


Is the relation y = x2 – 2x a function?

Solution

Each element of the domain (the x-values) would produce only one element of the range (the y-values), it is a function.

Example 3


Is the relation x2 – y2 = 9 a function?

Solution

Each element of the domain (the x-values) would correspond with 2 different values of the range (both a positive and negative y-value), the relation is NOT a function.

Example 4


Vertical Line TestIf no vertical line can be drawn so that it intersects a graph more than once, the graph is the graph of a function.


Use the vertical line test to determine whether the graph to the right is the graph of a function.

x

y

Yes, this is the graph of a function since vertical line will intersect this graph more than once.

Example 5



x

y

Example 5

Yes, this is the graph of a function since vertical line will intersect this graph more than once.



No, this is not the graph of a function. Vertical lines can be drawn that intersect the graph in two points.

x

y

Example 5


Since the graph of a linear equation is a line, all linear equations are functions, except those whose graph is a vertical line

Note: An equation of the form y = c is a horizontal line and IS a function.

An equation of the form x = c is a vertical line and IS NOT a function.


Find the domain and range of the function graphed to the right.

x

y

Domain: 3 ≤ x ≤ 4

Domain

Range: 4 ≤ y ≤ 2

Range

Example 6


Find the domain and range of the function graphed to the right.

x

y

Domain: all real numbers

DomainRange: y ≥ – 2

Range

Example 6


Function Notation

To denote that y is a function of x, we can write

y = f(x) (Read “f of x”)

Function notation

This notation means that y is a function of x or that y depends on x. For this reason, y is called the dependent variable and x the independent variable.


If f(x) = x2 – 2x, find f(–3).

Solution

f(–3) = (–3)2 – 2(–3)

= 9 – (–6)

= 15

Example 7

f(x) = x2 – 2x


Given the graph of the following function, find each function value by inspecting the graph.

f(5) = 6x

y

f(x)

f(4) = 3

f(5) = 1

f(6) = –6

Example 8

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