Copyright © 2011 Pearson Education, Inc.

FunctionsSection 2.1

Functions and Graphs

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2.1

If the value of a variable y is determined by the value of another variable x, then y is a function of x. The phrase “is a function of” means “is determined by.” If there is more than one value for y corresponding to a

particular x-value, then y is not determined by x and y is not a function of x.

When discussing two related variables, we can identify one as the first variable and the other as the second variable and consider the set of ordered pairs containing their corresponding values.

If the set of ordered pairs satisfies the function definition, then we say that the second variable is a function of the first.

The Function Concept

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2.1

Definition: FunctionA function is a rule that assigns each element in one set to

a unique element in a second set.

Definition: FunctionA function is a set of ordered pairs in which no two ordered

pairs have the same first coordinate and different second

coordinates.

The Function Concept

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Any set of ordered pairs is called a relation. A relation can be indicated by a verbal description, a

graph, a formula or equation, or a table, but there is always an underlying set of ordered pairs.

Not every relation is a function. A function is a special relation.

In general, if there is a vertical line that crosses a graph more than once, the graph is not the graph of a function. This criterion is known as the vertical line test.

Theorem: The Vertical Line TestA graph is the graph of a function if and only if there is no vertical line that crosses the graph more than once.

Identifying Functions

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

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

Domain and Range

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A function defined by a set of ordered pairs can be named with a letter. For example, f = {(2, 5), (3, 8)}. Since the function f pairs 2 with 5, we write f(2) = 5,

which is read as “the value of f at 2 is 5.” We also have f(3) = 8.

A function defined by an equation can also be named with a letter. We can use g(x), read as “g of x” as a symbol for the

second coordinate when the first coordinate is x. This notation is called function notation. The x in this notation is called a dummy variable

because the letter used is unimportant.

Function Notation

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Definition: Average Rate of Change from x1 to x2

If (x1, y1) and (x2, y2) are two ordered pairs of a function, we

define the average rate of change of the function as x

varies from x1 to x2, as the change in y-coordinates divided

by the change in x-coordinates (Δy/Δx) or

Definition: Difference Quotient

The difference quotient is the expression

.12

12

xxyy

.

h

xfhxf

The Average Rate ofChange of a Function

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2.1

The area of any square is given by A = s2. By the Pythagorean theorem, d2 = s2 + s2, d2

= 2s2, or s2 = d2/2. Since A = s2 and s2 = d2/2, we get the

formula

expressing the area of the square as a function of the length of the diagonal.

,2

2dA

s

s

d

We can find a formula for, or construct, a function relating two variables in a geometric figure.

For example, given that a square has diagonal of length d and side length s, write the area A as a function of the length of the diagonal.

Constructing Functions