Functions and Their Graphs

SECTION
3.2

Graphs of Functions; Piecewise-Defined Functions; Increasing and Decreasing Functions; Average Rate of Change

SKILLS OBJECTIVES


■	Classify functions as even, odd, or neither.

■	Determine whether functions are increasing, decreasing, or constant.

■	Calculate the average rate of change of a function.

■	Evaluate the difference quotient for a function.

■	Graph piecewise-defined functions.

CONCEPTUAL OBJECTIVES

■

Identify common functions.

■

Develop and graph piecewise-defined functions.


●	Identify and graph points of discontinuity.

●	State the domain and range.

■

Understand that even functions have graphs that are symmetric about the

■

Understand that odd functions have graphs that are symmetric about the origin.

Recognizing and Classifying Functions

3.2.1.1 Common Functions

Point-plotting techniques were introduced in Section 2.2, and we noted there that we would explore some more efficient ways of graphing functions in Chapter 3. The nine main functions you will read about in this section will constitute a “library” of functions that you should commit to memory. We will draw on this library of functions in the next section when graphing transformations are discussed. Several of these functions have been shown previously in this chapter, but now we will classify them specifically by name and identify properties that each function exhibits.

In Section 2.3, we discussed equations and graphs of lines. All lines (with the exception of vertical lines) pass the vertical line test, and hence are classified as functions. Instead of the traditional notation of a line,

, we use function notation and classify a function whose graph is a line as a linear function.

LINEAR FUNCTION

and

are real numbers.

The domain of a linear function

is the set of all real numbers

. The graph of this function has slope

and

Linear Function:	Slope:

One special case of the linear function is the constant function

CONSTANT FUNCTION

is any real number.

The graph of a constant function

is a horizontal line. The

corresponds to the point

. The domain of a constant function is the set of all real numbers

. The range, however, is a single value

. In other words, all

correspond to a single

Points that lie on the graph of a constant function

are

Domain:

Range:

Another specific example of a linear function is the function having a slope of one

and a

of zero

. This special case is called the identity function.

IDENTITY FUNCTION

The graph of the identity function has the following properties: It passes through the origin, and every point that lies on the line has equal

and

. Both the domain and the range of the identity function are the set of all real numbers

A function that squares the input is called the square function.

SQUARE FUNCTION

The graph of the square function is called a parabola and will be discussed in further detail in Chapters 4 and 8. The domain of the square function is the set of all real numbers

. Because squaring a real number always yields a positive number or zero, the range of the square function is the set of all nonnegative numbers. Note that the intercept is the origin and the square function is symmetric about the

. This graph is contained in quadrants I and II.

A function that cubes the input is called the cube function.

CUBE FUNCTION

The domain of the cube function is the set of all real numbers

. Because cubing a negative number yields a negative number, cubing a positive number yields a positive number, and cubing 0 yields 0, the range of the cube function is also the set of all real numbers

. Note that the only intercept is the origin and the cube function is symmetric about the origin. This graph extends only into quadrants I and III.

The next two functions are counterparts of the previous two functions: square root and cube root. When a function takes the square root of the input or the cube root of the input, the function is called the square root function or the cube root function, respectively.

SQUARE ROOT FUNCTION

In Section 3.1, we found the domain to be

. The output of the function will be all real numbers greater than or equal to zero. Therefore, the range of the square root function is

. The graph of this function will be contained in quadrant I.

CUBE ROOT FUNCTION

In Section 3.1, we stated the domain of the cube root function to be

. We see by the graph that the range is also

. This graph is contained in quadrants I and III and passes through the origin. This function is symmetric about the origin.

In Section 1.7, you read about absolute value equations and inequalities. Now we shift our focus to the graph of the absolute value function.

ABSOLUTE VALUE FUNCTION

Some points that are on the graph of the absolute value function are

, and

. The domain of the absolute value function is the set of all real numbers

, yet the range is the set of nonnegative real numbers. The graph of this function is symmetric with respect to the

and is contained in quadrants I and II.

A function whose output is the reciprocal of its input is called the reciprocal function.

RECIPROCAL FUNCTION

The only restriction on the domain of the reciprocal function is that

. Therefore, we say the domain is the set of all real numbers excluding zero. The graph of the reciprocal function illustrates that its range is also the set of all real numbers except zero. Note that the reciprocal function is symmetric with respect to the origin and is contained in quadrants I and III.

3.2.1.2 Even and Odd Functions

Of the nine functions discussed above, several have similar properties of symmetry. The constant function, square function, and absolute value function are all symmetric with respect to the

. The identity function, cube function, cube root function, and reciprocal function are all symmetric with respect to the origin. The term even is used to describe functions that are symmetric with respect to the

, or vertical axis, and the term odd is used to describe functions that are symmetric with respect to the origin. Recall from Section 2.2 that symmetry can be determined both graphically and algebraically. The box below summarizes the graphic and algebraic characteristics of even and odd functions.

EVEN AND ODD FUNCTIONS

Function	Symmetric with Respect to	On Replacing with
Even	or vertical axis
Odd	origin

The algebraic method for determining symmetry with respect to the

, or vertical axis, is to substitute

for

. If the result is an equivalent equation, the function is symmetric with respect to the

. Some examples of even functions are

; and

In any of these equations, if

is substituted for

, the result is the same; that is,

. Also note that, with the exception of the absolute value function, these examples are all even-degree polynomial equations. All constant functions are degree zero and are even functions.

The algebraic method for determining symmetry with respect to the origin is to substitute

for

. If the result is the negative of the original function, that is, if

, then the function is symmetric with respect to the origin and, hence, classified as an odd function. Examples of odd functions are

, and

. In any of these functions, if

is substituted for

, the result is the negative of the original function. Note that with the exception of the cube root function, these equations are odd-degree polynomials.

Be careful, though, because functions that are combinations of even- and odd-degree polynomials can turn out to be neither even nor odd, as we will see in Example 1.

EXAMPLE 1

Determining Whether a Function Is Even, Odd, or Neither

Determine whether the functions are even, odd, or neither.

(a)

(b)

(c)

Solution

(a)

Original function.
Replace with .
Simplify.
Because , we say that .

(b)

Original function.
Replace with .
Simplify.
Because , we say that .

(c)

Original function.
Replace with .
Simplify.
therefore the function is .

In parts (a), (b), and (c), we classified these functions as either even, odd, or neither, using the algebraic test. Look back at them now and reflect on whether these classifications agree with your intuition. In part (a), we combined two functions: the square function and the constant function. Both of these functions are even, and adding even functions yields another even function. In part (b), we combined two odd functions: the fifth-power function and the cube function. Both of these functions are odd, and adding two odd functions yields another odd function. In part (c), we combined two functions: the square function and the identity function. The square function is even, and the identity function is odd. In this part, combining an even function with an odd function yields a function that is neither even nor odd and, hence, has no symmetry with respect to the vertical axis or the origin.

■

YOUR TURN

Classify the functions as even, odd, or neither.

(a)

(b)

Increasing and Decreasing Functions

Look at the figure in the margin below. Graphs are read from left to right.

If we start at the left side of the graph and trace the red curve with our pen, we see that the function values (values in the vertical direction) are decreasing until arriving at the point

. Then, the function values increase until arriving at the point

. The values then remain constant

between the points

and

. Proceeding beyond the point

, the function values decrease again until the point

. Beyond the point

, the function values increase again until the point

. Finally, the function values decrease and continue to do so.

When specifying a function as increasing, decreasing, or constant, the intervals are classified according to the

For instance, in this graph, we say the function is increasing when

is between

and

and again when

is between

and

. The graph is classified as decreasing when

is less than

and again when

is between 0 and 2 and again when

is greater than 6. The graph is classified as constant when

is between

and 0. In interval notation, this is summarized as

Decreasing	Increasing	Constant

An algebraic test for determining whether a function is increasing, decreasing, or constant is to compare the value

of the function for particular points in the intervals.

INCREASING, DECREASING, AND CONSTANT FUNCTIONS


1.	A function is increasing on an open interval if for any and in , where , then .

2.	A function is decreasing on an open interval if for any and in , where , then .

3.	A function is constant on an open interval if for any and in , then .

In addition to classifying a function as increasing, decreasing, or constant, we can also determine the domain and range of a function by inspecting its graph from left to right:


■	The domain is the set of all (from left to right) where the function is defined.

■	The range is the set of all (from bottom to top) that the graph of the function corresponds to.

■	A solid dot on the left or right end of a graph indicates that the graph terminates there and the point is included in the graph.

■	An open dot indicates that the graph terminates there and the point is not included in the graph.

■	Unless a dot is present, it is assumed that a graph continues indefinitely in the same direction. (An arrow is used in some books to indicate direction.)

EXAMPLE 2

Finding Intervals When a Function Is Increasing or Decreasing

Given the graph of a function:

(a)	State the domain and range of the function.

(b)	Find the intervals when the function is increasing, decreasing, or constant.

Solution

(a)

(b)

Reading the graph from left to right, we see that the graph


■	decreases from the point to the point .

■	is constant from the point to the point .

■	decreases from the point to the point .

■	increases from the point on.

The intervals of increasing and decreasing correspond to the

We say that this function is

Note: The intervals of increasing or decreasing are defined on open intervals. This should not be confused with the domain. For example, the point

is included in the domain of the function but not in the interval where the function is classified as decreasing.

Average Rate of Change

How do we know how much a function is increasing or decreasing? For example, is the price of a stock slightly increasing or is it doubling every week? One way we determine how much a function is increasing or decreasing is by calculating its average rate of change.

Let

and

be two points that lie on the graph of a function

. Draw the line that passes through these two points

and

. This line is called a secant line.

Note that the slope of the secant line is given by

, and recall that the slope of a line is the rate of change of that line. The slope of the secant line is used to represent the average rate of change of the function.

AVERAGE RATE OF CHANGE

Let

and

be two distinct points,

, on the graph of the function

. The average rate of change of

between

and

is given by

EXAMPLE 3

Average Rate of Change

Find the average rate of change of

from:

(a)

(b)

(c)

Solution

(a)

Write the average rate of change formula.
Let and .
Substitute and .
Simplify.

(b)

Write the average rate of change formula.
Let and .
Substitute and .
Simplify.

(c)

Write the average rate of change formula.
Let and .
Substitute and .
Simplify.

Graphical Interpretation: Slope of the Secant Line

(a)

Average rate of change of

from

: Decreasing at a rate of 1

(b)

Average rate of change of

from

: Increasing at a rate of 1

(c)

Average rate of change of

from

: Increasing at a rate of 15

■

YOUR TURN

Find the average rate of change of

from:

(a)

(b)

The average rate of change can also be written in terms of the difference quotient.

When written in this form, the average rate of change is called the difference quotient.

DEFINITION

Difference Quotient

The expression

, where

, is called the difference quotient.

The difference quotient is more meaningful when

is small. In calculus the difference quotient is used to define a derivative.

EXAMPLE 4

Calculating the Difference Quotient

Calculate the difference quotient for the function

Solution

■

YOUR TURN

Calculate the difference quotient for the function

Piecewise-Defined Functions

Most of the functions that we have seen in this text are functions defined by polynomials. Sometimes the need arises to define functions in terms of pieces. For example, most plumbers charge a flat fee for a house call and then an additional hourly rate for the job. For instance, if a particular plumber charges

to drive out to your house and work for 1 hour and then an additional

an hour for every additional hour he or she works on your job, we would define this function in pieces. If we let

be the number of hours worked, then the charge is defined as

If we were to graph this function, we would see that there is 1 hour that is constant and after that the function continually increases.

Another piecewise-defined function is the absolute value function. The absolute value function can be thought of as two pieces: the line

(when

is negative) and the line

(when

is nonnegative). We start by graphing these two lines on the same graph.

The absolute value function behaves like the line

when

is negative (erase the blue graph in quadrant IV) and like the line

when

is positive (erase the red graph in quadrant III).

The next example is a piecewise-defined function given in terms of functions in our “library of functions.” Because the function is defined in terms of pieces of other functions, we draw the graph of each individual function, and then for each function, darken the piece corresponding to its part of the domain. This is like the procedure above for the absolute value function.

EXAMPLE 5

Graphing Piecewise-Defined Functions

Graph the piecewise-defined function, and state the domain, range, and intervals when the function is increasing, decreasing, or constant.

Solution

Graph each of the functions on the same plane.

Square function:

Constant function:

Identity function:

The points to focus on in particular are the

where the pieces change over—that is,

and

Let's now investigate each piece. When

, this function is defined by the square function,

, so darken that particular function to the left of

. When

, the function is defined by the constant function,

, so darken that particular function between the

values of

and 1. When

, the function is defined by the identity function,

, so darken that function to the right of

. Erase everything that is not darkened, and the resulting graph of the piecewise-defined function is given on the right. This function is defined for all real values of

, so the domain of this function is the set of all real numbers. The values that this function yields in the vertical direction are all real numbers greater than or equal to 1. Hence, the range of this function is

. The intervals of increasing, decreasing, and constant are as follows:

The term continuous implies that there are no holes or jumps and that the graph can be drawn without picking up your pencil. A function that does have holes or jumps and cannot be drawn in one motion without picking up your pencil is classified as discontinuous, and the points where the holes or jumps occur are called points of discontinuity.

The previous example illustrates a continuous piecewise-defined function. At the

junction, the square function and constant function both pass through the point

. At the

junction, the constant function and the identity function both pass through the point

. Since the graph of this piecewise-defined function has no holes or jumps, we classify it as a continuous function.

The next example illustrates a discontinuous piecewise-defined function.

EXAMPLE 6

Graphing a Discontinuous Piecewise-Defined Function

Graph the piecewise-defined function, and state the intervals where the function is increasing, decreasing, or constant, along with the domain and range.

Solution

Graph these functions on the same plane.

Linear function:

Identity function:

Constant function:

Darken the piecewise-defined function on the graph. For all values less than zero

the function is defined by the linear function. Note the use of an open circle, indicating up to but not including

. For values

, the function is defined by the identity function.

The circle is filled in at the left endpoint,

. An open circle is used at

. For all values greater than 2,

, the function is defined by the constant function. Because this interval does not include the point

, an open circle is used.

At what intervals is the function increasing, decreasing, or constant? Remember that the intervals correspond to the

Decreasing:

Increasing:

Constant:

The function is defined for all values of

except

The output of this function (vertical direction) takes on the

and the additional single value

We mentioned earlier that a discontinuous function has a graph that exhibits holes or jumps. In this example, the point

corresponds to a jump, because you would have to pick up your pencil to continue drawing the graph. The point

corresponds to both a hole and a jump. The hole indicates that the function is not defined at that point, and there is still a jump because the identity function and the constant function do not meet at the same

■

YOUR TURN

Graph the piecewise-defined function, and state the intervals where the function is increasing, decreasing, or constant, along with the domain and range.

Piecewise-defined functions whose “pieces” are constants are called step functions. The reason for this name is that the graph of a step function looks like steps of a staircase. A common step function used in engineering is the Heaviside step function (also called the unit step function):

This function is used in signal processing to represent a signal that turns on at some time and stays on indefinitely.

A common step function used in business applications is the greatest integer function.

GREATEST INTEGER FUNCTION

	1.0	1.3	1.5	1.7	1.9	2.0
	1	1	1	1	1	2

SECTION
3.2

SUMMARY

Name	Range	Even/Odd
Linear		Neither (unless )
Constant	or	Even
Identity		Odd
Square		Even
Cube		Odd
Square Root		Neither
Cube Root		Odd
Absolute Value		Even
Reciprocal		Odd

Domain and Range of a Function


■	Implied domain: Exclude any values that lead to the function being undefined (dividing by zero) or imaginary outputs (square root of a negative real number).

■	Inspect the graph to determine the set of all inputs (domain) and the set of all outputs (range).

Finding Intervals Where a Function Is Increasing, Decreasing, or Constant


■	Increasing: Graph of function rises from left to right.

■	Decreasing: Graph of function falls from left to right.

■	Constant: Graph of function does not change height from left to right.

Average Rate of Change
Difference Quotient

Piecewise-Defined Functions


■	Continuous: You can draw the graph of a function without picking up the pencil.

■	Discontinuous: Graph has holes and/or jumps.

SECTION
3.2

EXERCISES

SKILLS

In Exercises 1-24, determine whether the function is even, odd, or neither.

1.	neither So, not even. So, not odd. Thus, .

3.	even So, . Thus, cannot be odd.

5.	odd So, not even. So, .

7.	neither So, not even. So, not odd. Thus, .

9.	odd So, not even. So, .

10.

11.	even So, . Thus, cannot be odd.

12.

13.	even So, . Thus, cannot be odd.

14.

15.	neither So, not even. So, not odd. Thus, .

16.

17.	neither So, not even. So, not odd. Thus, .

18.

19.	neither So, not even. So, not odd. Thus, .

20.

21.	neither So, not even. So, not odd. Thus, .

22.

23.

24.

In Exercises 25-36, state the (a) domain, (b) range, and (c)

-interval(s) where the function is increasing, decreasing, or constant. Find the values of (d)

, (e)

, and (f)

25.

26.

27.

28.

29.

30.

31.

32.

33.

34.

35.

36.

In Exercises 37-44, find the difference quotient

for each function.

37.

38.

39.

40.

41.

42.

43.

44.

In Exercises 45-52, find the average rate of change of the function from

45.

46.

47.

48.

49.

50.

51.

52.

In Exercises 53-78, graph the piecewise-defined functions. State the domain and range in interval notation. Determine the intervals where the function is increasing, decreasing, or constant.

53.

54.

55.

56.

57.

58.

59.

60.

61.

62.

63.

64.

65.

66.

67.

68.

69.

70.

71.

72.

73.

74.

75.

76.

77.

78.

APPLICATIONS

For Exercises 79 and 80, refer to the following:

A manufacturer determines that his profit and cost functions over one year are represented by the following graphs.

79.

Business.

Find the intervals on which profit is increasing, decreasing, and constant.

80.	Business. Find the intervals on which cost is increasing, decreasing, and constant.

81.

Budget: Costs.

The Kappa Kappa Gamma sorority decides to order custom-made T-shirts for its Kappa Krush mixer with the Sigma Alpha Epsilon fraternity. If the sorority orders 50 or fewer T-shirts, the cost is

per shirt. If it orders more than 50 but less than or equal to 100, the cost is

per shirt. If it orders more than 100, the cost is

per shirt. Find the cost function

as a function of the number of T-shirts

ordered.

82.

Budget: Costs.

The marching band at a university is ordering some additional uniforms to replace existing uniforms that are worn out. If the band orders 50 or fewer, the cost is

per uniform. If it orders more than 50 but less than or equal to 100, the cost is

per uniform. Find the cost function

as a function of the number of new uniforms

ordered.

83.

Budget: Costs.

The Richmond rowing club is planning to enter the Head of the Charles race in Boston and is trying to figure out how much money to raise. The entry fee is

per boat for the first 10 boats and

for each additional boat. Find the cost function

as a function of the number of boats

the club enters.

84.	Phone Cost: Long-Distance Calling. A phone company charges per minute for the first 10 minutes of an international long-distance phone call and per minute every minute after that. Find the cost function as a function of the length of the phone call in minutes.

85.

Event Planning.

A young couple are planning their wedding reception at a yacht club. The yacht club charges a flat rate of

to reserve the dining room for a private party. The cost of food is

per person for the first 100 people and

per person for every additional person beyond the first 100. Write the cost function

as a function of the number of people

attending the reception.

86.	Home Improvement. An irrigation company gives you an estimate for an eight-zone sprinkler system. The parts are , and the labor is per hour. Write a function that determines the cost of a new sprinkler system if you choose this irrigation company.

87.

Sales.

A famous author negotiates with her publisher the monies she will receive for her next suspense novel. She will receive

up front and a

royalty rate on the first 100,000 books sold, and

on any books sold beyond that. If the book sells for

and royalties are based on the selling price, write a royalties function

as a function of total number

of books sold.

88.	Sales. Rework Exercise 87 if the author receives up front, for the first 100,000 books sold, and on any books sold beyond that.

89.

Profit.

Some artists are trying to decide whether they will make a profit if they set up a Web-based business to market and sell stained glass that they make. The costs associated with this business are

per month for the website and

per month for the studio they rent. The materials cost

for each work in stained glass, and the artists charge

for each unit they sell. Write the monthly profit as a function of the number of stained-glass units they sell.

90.

Profit.

Philip decides to host a shrimp boil at his house as a fundraiser for his daughter's AAU basketball team. He orders gulf shrimp to be flown in from New Orleans. The shrimp costs

per pound. The shipping costs

. If he charges

per person, write a function

that represents either his loss or profit as a function of the number of people

that attend. Assume that each person will eat 1 pound of shrimp.

91.

Postage Rates.

The following table corresponds to first-class postage rates for the U.S. Postal Service. Write a piecewise-defined function in terms of the greatest integer function that models this cost of mailing flat envelopes first class.

Weight Less Than (ounces)	First-Class Rate (Flat Envelopes)
1	$0.80
2	$0.97
3	$1.14
4	$1.31
5	$1.48
6	$1.65
7	$1.82
8	$1.99
9	$2.16
10	$2.33
11	$2.50
12	$2.67
13	$2.84

92.

Postage Rates.

Weight Less Than (ounces)	First-Class Rate (Parcels)
1	$1.13
2	$1.30
3	$1.47
4	$1.64
5	$1.81
6	$1.98
7	$2.15
8	$2.32
9	$2.49
10	$2.66
11	$2.83
12	$3.00
13	$3.17

A square wave is a waveform used in electronic circuit testing and signal processing. A square wave alternates regularly and instantaneously between two levels.

93.

Electronics: Signals.

Write a step function

that represents the following square wave.

94.

Electronics: Signals.

Write a step function

that represents the following square wave, where

represents frequency in Hz.

For Exercises 95 and 96, refer to the following table:

Global Carbon Emissions from Fossil Fuel Burning

Year	Million of Tons of carbon
1900	500
1925	1000
1950	1500
1975	5000
2000	7000

95.

Climate Change: Global Warming.

What is the average rate of change in global carbon emissions from fossil fuel burning from

(a)	1900 to 1950? 20 million tons per year

(b)	1950 to 2000? 110 million tons per year

96.

Climate Change: Global Warming.

What is the average rate of change in global carbon emissions from fossil fuel burning from

(a)	1950 to 1975?

(b)	1975 to 2000?

For Exercises 97 and 98, use the following information:

The height (in feet) of a falling object with an initial velocity of

per second launched straight upward from the ground is given by

, where

is time (in seconds).

97.	Falling Objects. What is the average rate of change of the height as a function of time from to ? 0 feet per second

98.	Falling Objects. What is the average rate of change of the height as a function of time from to ?

For Exercises 99 and 100, refer to the following:

An analysis of sales indicates that demand for a product during a calendar year (no leap year) is modeled by

where

is demand in thousands of units and

is the day of the year and

represents January 1.

99.

Economics.

Find the average rate of change of the demand of the product over the first quarter.

100.	Economics. Find the average rate of change of the demand of the product over the fourth quarter.

CATCH THE MISTAKE

In Exercises 101-104, explain the mistake that is made.

101.

Graph the piecewise-defined function. State the domain and range.

Solution:

Draw the graphs of

and

Darken the function

when

and the function

when

. This gives us the familiar absolute value graph.

Domain:

Range:

This is incorrect. What mistake was made?

102.

Graph the piecewise-defined function. State the domain and range.

Solution:

Draw the graphs of

and

Darken the function

when

and the function

when

The resulting graph is as shown.

Domain:

Range:

This is incorrect. What mistake was made?

103.	The cost of airport Internet access is for the first 30 minutes and per minute for each additional minute. Write a function describing the cost of the service as a function of minutes used online. Solution: This is incorrect. What mistake was made? The portion of for should be:

104.

Most money market accounts pay a higher interest with a higher principal. If the credit union is offering

on accounts with less than or equal to

and

on the additional money over

, write the interest function

that represents the interest earned on an account as a function of dollars in the account.

Solution:

This is incorrect. What mistake was made?

CONCEPTUAL

In Exercises 105-108, determine whether each statement is true or false.

105.	The identity function is a special case of the linear function. true True. This corresponds to with .

106.	The constant function is a special case of the linear function.

107.	If an odd function has an interval where the function is increasing, then it also has to have an interval where the function is decreasing. false False. For instance, is always increasing.

108.	If an even function has an interval where the function is increasing, then it also has to have an interval where the function is decreasing.

CHALLENGE

In Exercises 109 and 110, for

and

real numbers, can the function given ever be a continuous function? If so, specify the value for

and

that would make it so.

109.	yes, if The individual pieces used to form , namely , are continuous on . So, the only with which we need to be concerned regarding the continuity of is . For to be continuous at 2, we need , which is the same as .