Chapter 3 Section 5 Copyright © 2011 Pearson Education, Inc.

Copyright © 2011 Pearson Education, Inc.

Chapter 3Chapter 3Section 5Section 5

Copyright © 2011 Pearson Education, Inc.

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Introduction to Functions

Distinguish between independent and dependent variables.Define and identify relations and functions.Find the domain and range.Identify functions defined by graphs and equations.Use function notation.Graph linear and constant functions.

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3.53.53.53.5

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Copyright © 2011 Pearson Education, Inc. Slide 3.5- 3

1Objective

Distinguish between independent and dependent variables.


We often describe one quantity in terms of another:The amount of your paycheck if you are paid hourly depends on the number of hours you worked.

The cost at the gas station depends on the number of gallons of gas you pumped into your car.

The distance traveled by a car moving at a constant speed depends on the time traveled.

If the value of the variable y depends on the value of the variable x, then y is the dependent variable and x is the independent variable.


Objective 2

Define and identify relations and functions.


(x, y)

Dependent VariableIndependent Variable

Relation

A relation is a set of ordered pairs.

Function

A function is a relation in which, for each value of the first component of the ordered pairs, there is exactly one value of the second component.


EXAMPLE 1

Determine whether each relation defines a function.

a. {(0, 3), (–1, 2), (–1, 3)}

No, the same x-value is paired with a different y-value.

In a function, no two ordered pairs can have the same first component and different second components.

b. {(5, 4), (6, 4), (7, 4)}

Yes, each different x-value is paired with a y-value. This does not violate the definition of a function.


Relations and functions can also be expressed as a correspondence or mapping from one set to another.

1

2

3

2

5

6

1

2

2

6

Function

Not a function


Objective 3

Find the domain and range.


Domain and Range

In a relation, the set of all values of the independent variable (x) is the domain.

The set of all values of the dependent variable (y) is the range.


EXAMPLE 2

Give the domain and range of the relation represented by the table for cellular telephone subscribers. Does it define a function?

Year Subscribers

1999 86,047

2000 109,478

2001 128,375

2002 140,767

2003 158,722

2004 182,140

Domain: {1999, 2000, 2001, 2002, 2003, 2004}

Range: {86,047, 109,478, 128,375, 140,767, 158,722, 182,140}

Yes; it is a function.


EXAMPLE 3

Give the domain and range of the relation.

Range

Range: (, 4]

Domain: (, )

The arrowheads indicate that the line extends indefinitely left and right.


Agreement on Domain

Unless specified otherwise, the domain of a relation is assumed to be all real numbers that produce real numbers when substituted for the independent variable.


Objective 4

Identify functions defined by graphs and equations.


Vertical Line Test

If every vertical line intersects the graph of a relation in no more than one point, then the relation is a function.


Function Not a Function


EXAMPLE 4

Use the vertical line test to decide whether the relation shown below is a function.

Yes, the relation is a function.


EXAMPLE 5

Decide whether each equation defines y as a function of x, and give the domain.

a. y = –2x + 7

y is always found by multiplying by negative two and adding 7. Each value of x corresponds to just one value of y.

b.

Yes, (, )

5 6y x

6,

5 Yes,

5 6 0x 5 6x

6

5x


continued

c. y4 = x

d. y 4x + 2

e.6

5 3y

x

No, [0, )

No, (, )

Yes5 5

, ,3 3

4 2 0x 4 2x

1

2x

0 5 3x 5 3x 5

3x

The denominator would be zero and this is undefined so it is not included in the domain.


Variations of the Definition of a Function

1. A function is a relation in which, for each value of the first component of the ordered pairs, there is exactly one value of the second component.

2. A function is a set of distinct ordered pairs in which no first component is repeated.

3. A function is a rule or correspondence that assigns exactly one range value to each domain value.


Objective 5

Use function notation.


When a function f is defined with a rule or an equation using x and y for the independent and dependent variables, we say, “y is a function of x” to emphasize that y depends on x. We use the notation

y = f(x),

called function notation, to express this and read f(x) as “f of x.”

y = f(x) = 9x – 5

Name of the function

Name of the independent variable

Value of the function

Defining expression


EXAMPLE 6

Let Find the following

a. f(–3) b. f(t)

3 5( ) .

2

xf x

3 5( )

2

xf x

3( ) 5(

3)3

2f

9 5

2

7

3 5( )

2

xf x

3( ) 5( )

2f

tt


EXAMPLE 7

Let g(x) = 5x – 1. Find and simplify g(m + 2).

g(x) = 5x – 1

g(m + 2) = 5(m + 2) – 1

= 5m + 10 – 1

= 5m + 9


EXAMPLE 8

Find f(2) for each function.

a. b. f = {(2, 6), (4, 2)}x f(x)

–4 16

–2 4

0 0

2 4

4 16

f(2) = 4

f(2) = 6


continued

c. f(x) = –x2 d. The function graphed.

f(2) = –22

f(2) = 4

f(2) = 3


Finding an Expression for f(x)

Step 1 Solve the equation for y.

Step 2 Replace y with f(x).


EXAMPLE 9

Rewrite the equation using function notation f(x). Then find f(1) and f(a).

x2 – 4y = 3

Step 1: Solve for y. 24 3y x 2 3

4 4

xy

2 3

4 4y

x

2

(4

)3

4

xf x


continued

f(1) f(a)2

4(1)

3

4

xf

21( )( )

41

3

4f

1 3 1

4 4 2

2

(4

)3

4

xf a

2

(4

)( ) 3

4

af a

2 3

4

a


Objective 6

Graph linear and constant functions.


Linear Function

A function that can be defined by

f(x) = ax + b

for real numbers a and b is a linear function. The value of a is the slope m of the graph of the function.


EXAMPLE 10

Graph 1 3

( )2 2

f x x

1

2( )

3

2f x x

y-interceptSlope

Chapter 3 Section 5 Copyright © 2011 Pearson Education, Inc.

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