Section 2.6 A Library of Functions

Before Starting this Section, Review

  1. 1 Equation of a line (Section 2.3 )

  2. 2 Absolute value (Section P.1 , page 10)

  3. 3 Graph of an equation (Section 2.2 , page 186)

  4. 4 Rational exponents (Section P.6 , page 68)

Objectives

  1. 1 Relate linear functions to linear equations.

  2. 2 Graph square root and cube root functions.

  3. 3 Evaluate and graph piecewise functions.

  4. 4 Graph additional basic functions.

The Megatooth Shark

The giant “Megatooth” shark (Carcharodon megalodon), once estimated to be 100 to 120 feet in length, is the largest meat-eating fish that ever lived. Its actual length has been the subject of scientific controversy. Sharks first appeared in the oceans more than 400 million years ago, almost 200 million years before the dinosaurs. A shark’s skeleton is made of cartilage, which decomposes rather quickly, making complete shark fossils rare. Consequently, scientists rely on calcified vertebrae, fossilized teeth, and small skin scales to reconstruct the evolutionary record of sharks.

The great white shark is the closest living relative to the giant “Megatooth” shark and has been used as a model to reconstruct the “Megatooth.” John Maisey, curator at the American Museum of Natural History in New York City, used a partial set of “Megatooth” teeth to make a comparison with the jaws of the great white shark. In Example 2, we use a formula to calculate the length of the “Megatooth” on the basis of the height of the tooth of the largest known “Megatooth” specimen.

Linear Functions

  1. 1 Relate linear functions to linear equations.

We know from Section 2.3 that the graph of a linear equation y=mx+by=mx+b is a straight line with slope m and y-intercept b. A linear function has a similar definition.

The domain of a linear function is the interval (−∞, ∞)(−∞, ∞) because the expression mx+bmx+b is defined for any real number x. Figure 2.69 shows that the graph extends indefinitely to the left and right. The range of a nonconstant linear function also is the interval (−∞, ∞)(−∞, ∞) because the graph extends indefinitely upward and downward. The range of a constant function f(x)=bf(x)=b is the single real number b. See Figure 2.69(c).

Example 1 Writing a Linear Function

Write a linear function g for which g(1)=4g(1)=4 and g(−3)=−2.g(−3)=−2.

Solution

We need to find the equation of the line passing through the two points

(x1, y1)=(1, 4) and (x2, y2)=(−3,−2).
(x1, y1)=(1, 4) and (x2, y2)=(−3,−2).

The slope of the line is given by

m=y2−y1x2−x1=−2−4−3−1=−6−4=32.
m=y2−y1x2−x1=−2−4−3−1=−6−4=32.

Next, we use the point–slope form of a line.

y−y1=m(x−x1)Point–slope form of a liney−4=32(x−1)Substitute x1=1, y1=4, and m=32.y−4=32x−32Distributive propertyy=32x+52Add 4=82 to both sides and simplify.g(x)=32x+52Function notation
y−y1y−4y−4yg(x)=====m(x−x1)32(x−1)32x−3232x+5232x+52Point–slope form of a lineSubstitute x1=1, y1=4, and m=32.Distributive propertyAdd 4=82 to both sides and simplify.Function notation

The graph of the function y=g(x)y=g(x) is shown in Figure 2.70.

Figure 2.70

Graph of y=g(x)y=g(x)

Practice Problem 1

  1. Write a linear function g for which g(−2)=2g(−2)=2 and g(1)=8.g(1)=8.

Example 2 Determining the Length of the “Megatooth” Shark

The largest known “Megatooth” specimen is a tooth that has a total height of 15.6 centimeters. Calculate the length of the “Megatooth” shark from which it came by using the formula

shark length =(0.96)(height of tooth)−0.22,
shark length =(0.96)(height of tooth)−0.22,

where shark length is measured in meters and tooth height is measured in centimeters.

Solution

Shark length =(0.96)(height of tooth)−0.22FormulaShark length =(0.96)(15.6)−0.22Replace height oftooth with 15.6.=14.756Simplify.
Shark length Shark length ===(0.96)(height of tooth)−0.22(0.96)(15.6)−0.2214.756FormulaReplace height oftooth with 15.6.Simplify.

The shark’s length is approximately 14.8 meters (≈48.6 feet).(≈48.6 feet).

Practice Problem 2

  1. If the “Megatooth” specimen was a tooth measuring 16.4 centimeters, what was the length of the “Megatooth” shark from which it came?

Square Root and Cube Root Functions

  1. 2 Graph square root and cube root functions.

So far, we have learned to graph linear functions: y=mx+b;y=mx+b; the squaring function: y=x2;y=x2; and the cubing function: y=x3.y=x3. We now add some common functions to this list.

Square Root Function

Recall (page 61) that for every nonnegative real number x, there is only one principal square root denoted by √x.x−−√. Therefore, the equation y=√xy=x−−√ defines a function, called the square root function. The domain and range of the square root function is [0, ∞).[0, ∞).

Example 3 Graphing the Square Root Function

Graph f(x)=√x.f(x)=x−−√.

Solution

To sketch the graph of f(x)=√x,f(x)=x−−√, we make a table of values. For convenience, we have selected the values of x that are perfect squares. See Table 2.10. We then plot the ordered pairs (x, y) and draw a smooth curve through the plotted points. See Figure 2.71.

Figure 2.71 Graph of y=√xy=x−−√

TABLE 2.10

x y=f(x)y=f(x) (x, y)
0 √0=00–√=0 (0, 0)
1 √1=11–√=1 (1, 1)
4 √4=24–√=2 (4, 2)
9 √9=39–√=3 (9, 3)
16 √16=416−−√=4 (16, 4)

The domain of f(x)=√xf(x)=x−−√ is [0, ∞),[0, ∞), and its range also is [0, ∞).[0, ∞).

Practice Problem 3

  1. Graph g(x)=√−xg(x)=−x−−−√ and find its domain and range.

Cube Root Function

There is only one cube root for each real number x. Therefore, the equation y=3√xy=x−−√3 defines a function called the cube root function. Both the domain and range of the cube root function are (−∞, ∞).(−∞, ∞).

Example 4 Graphing the Cube Root Function

Graph f(x)=3√x.f(x)=x−−√3.

Solution

We use the point-plotting method to graph f(x)=3√x.f(x)=x−−√3. For convenience, we select values of x that are perfect cubes. See Table 2.11. The graph of f(x)=3√xf(x)=x−−√3 is shown in Figure 2.72.

Figure 2.72 Graph of y=3√xy=x−−√3

TABLE 2.11

x y=f(x)y=f(x) (x, y)
−8−8 3√−8=−2−8−−−√3=−2 (−8,−2)(−8,−2)
−1−1 3√−1=−1−1−−−√3=−1 (−1,−1)(−1,−1)
0 3√0=00–√3=0 (0, 0)
1 3√1=11–√3=1 (1, 1)
8 3√8=28–√3=2 (8, 2)

The domain of f(x)=3√xf(x)=x−−√3 is (−∞, ∞),(−∞, ∞), and its range also is (−∞, ∞).(−∞, ∞).

Practice Problem 4

  1. Graph g(x)=3√−xg(x)=−x−−−√3 and find its domain and range.

Piecewise Functions

  1. 3 Evaluate and graph piecewise functions.

In the definition of some functions, different rules for assigning output values are used on different parts of the domain. Such functions are called piecewise functions. One example of a piecewise function is a line graph where data points are connected by straight-line segments.

The graph in Figure 2.73 represents the relative number of news headlines for popular Hollywood figures over a period of one year. Line graphs are powerful tools to visualize data and to study trends.

Figure 2.73 Source: Google trends

In Example 8, we will see how such graphs are created using piecewise functions.

Practice Problem 5

  1. Let f(x)={x2if x≤−12xif x>−1f(x)={x22xif x≤−1if x>−1. Find f(−2)f(−2) and f(3).

Example 6 Finding and Evaluating a Piecewise Function

In Peach County, Georgia, a section of the interstate highway has a speed limit of 55 miles per hour (mph). If you are caught speeding between 56 and 74 mph, your fine is $50 plus $3 for every mile per hour over 55 mph. For 75 mph and higher, your fine is $150 plus $5 for every mile per hour over 75 mph.

  1. Find a piecewise function that gives your fine.

  2. What is the fine for driving 60 mph?

  3. What is the fine for driving 90 mph?

Solution

  1. Let f(x) be the piecewise function that represents your fine for speeding at x miles per hour. We express f(x) as a piecewise function:

    f(x)={50+3(x−55),56≤x<75150+5(x−75),x≥75.
    f(x)={50+3(x−55),150+5(x−75),56≤x<75x≥75.

  2. The first line of the function means that if 56≤x<75,56≤x<75, your fine is

    f(x)=50+3(x−55).
    f(x)=50+3(x−55).

    Suppose you are caught driving 60 mph. Then we have

    f(x)=50+3(x−55)Expression used for 56≤x<75f(60)=50+3(60−55)Substitute 60 for x.=65Simplify.
    f(x)f(60)===50+3(x−55)50+3(60−55)65Expression used for 56≤x<75Substitute 60 for x.Simplify.

    Your fine for speeding at 60 mph is $65.

  3. The second line of the function means that if x≥75,x≥75, your fine is f(x)=150+5(x−75).f(x)=150+5(x−75). Suppose you are caught driving 90 mph. Then we have

    f(x)=150+5(x−75)Expression used for x≥75f(90)=150+5(90−75)Substitute 90 for x.=225Simplify.
    f(x)f(90)===150+5(x−75)150+5(90−75)225Expression used for x≥75Substitute 90 for x.Simplify.

    Your fine for speeding at 90 mph is $225.

Practice Problem 6

  1. Repeat Example 6 if the speeding penalties are changed to $50 plus $4 for every mile per hour over 55 mph and $200 plus $5 for every mile per hour over 75 mph.

Example 7 Writing a Piecewise Function from the Set of Points

Construct the line graph from the data in Table 2.12 and express the function representing the line graph as a piecewise function.

TABLE 2.12

Time Value
1 1
3 5
5 2

Solution

Draw three points (1, 1), (3, 5), and (5, 2) and connect them with straight lines (see Figure  2.74). The graph of f is made up of two parts:

Figure 2.74

  1. A line segment passing through (1, 1) and (3, 5) over the interval [1, 3]. The slope of this line is

    m=ΔyΔx=5−13−1=42=2.
    m=ΔyΔx=5−13−1=42=2.

    Then

    y−y1=m(x−x1)Point–slope formy−y1=2(x−x1)Replace m with 2.y−1=2(x−1)Point (x1, y1)=(1, 1) is on the line;replace both x1 and y1 with 1.y−1=2x−2Distribute.y=2x−1Solve for y.
    y−y1y−y1y−1y−1y=====m(x−x1)2(x−x1)2(x−1)2x−22x−1Point–slope formReplace m with 2.Point (x1, y1)=(1, 1) is on the line;replace both x1 and y1 with 1.Distribute.Solve for y.

    So f(x)=2x−1f(x)=2x−1 for 1≤x≤31≤x≤3.

  2. A line segment passing through (3, 5) and (5, 2) over the interval [3, 5]. The slope of this line is

    m=ΔyΔx=2−55−3=−32=−32.
    m=ΔyΔx=2−55−3=−32=−32.

    Then

    y−y1=m(x−x1)Point–slope formy−y1=−32(x−x1)Replace m with −32.y−5=−32(x−3)Point (x1, y1)=(3, 5) is on the line;replace x1 with 3 and y1 with 5.y−5=−32x+92Distribute.y=−32x+192Solve for y.
    y−y1y−y1y−5y−5y=====m(x−x1)−32(x−x1)−32(x−3)−32x+92−32x+192Point–slope formReplace m with −32.Point (x1, y1)=(3, 5) is on the line;replace x1 with 3 and y1 with 5.Distribute.Solve for y.

    So f(x)=−32x+192f(x)=−32x+192 for 3≤x≤5.3≤x≤5.

    Observe that the two line segments that form the graph of this function are joined together without a gap, so we can attach the point (3, 5) to either part. Combining (a) and (b), respectively, we have

    g(x)={2x−1if 1≤x≤3−32x+192if 3<x≤5.
    g(x)=⎧⎩⎨2x−1−32x+192if 1≤x≤3if 3<x≤5.

Practice Problem 7

  1. Repeat Example 7 for Table 2.13 .

    TABLE 2.13

    Time Value
    1 5
    3 2
    5 4

Graphing Piecewise Functions

Let’s consider how to graph a piecewise function. The absolute value function, f(x)=|x|,f(x)=|x|, can be expressed as a piecewise function by using the definition of absolute value:

f(x)=|x|={−xif x<0xif x≥0.
f(x)=|x|={−xxif x<0if x≥0.

The first line in the function means that if x<0,x<0, we use the equation y=f(x)=−x.y=f(x)=−x. So, if x=−3,x=−3, then

y=f(−3)=−(−3)Substitute −3 for x in f(x)=−x.=3Simplify.
y=f(−3)==−(−3)3Substitute −3 for x in f(x)=−x.Simplify.

Thus, (−3, 3)(−3, 3) is a point on the graph of y=|x|.y=|x|. However, if x≥0,x≥0, we use the second line in the function, which is y=f(x)=x.y=f(x)=x. So, if x=2,x=2, then

y=f(2)=2Substitute 2 for x in f(x)=x
y=f(2)=2Substitute 2 for x in f(x)=x

So (2, 2) is a point on the graph of y=|x|.y=|x|. The two pieces y=−xy=−x and y=xy=x are linear functions. We graph the appropriate parts of these lines (y=−x(y=−x for x<0x<0 and y=xy=x for x≥0)x≥0) to form the graph of y=|x|.y=|x|. See Figure 2.75.

Figure 2.75 Graph of y=|x|y=|x|

Example 8 Graphing a Piecewise Function

Let

F(x)={−2x+1if x<13x+1if x≥1.
F(x)={−2x+13x+1if x<1if x≥1.

Sketch the graph of y=F(x).y=F(x).

Solution

In the definition of F, the formula changes at x=1.x=1. We call such numbers the transition points of the formula. For the function F, the only transition point is 1. Generally, to graph a piecewise function, we graph the function separately over the open intervals determined by the transition points and then graph the function at the transition points themselves. For the function y=F(x),y=F(x), we graph the equation y=−2x+1y=−2x+1 on the interval (−∞, 1).(−∞, 1). See Figure 2.76(a). Next, we graph the equation y=3x+1y=3x+1 on the interval (1, ∞)(1, ∞) and at the transition point 1, where y=F(1)=3(1)+1=4.y=F(1)=3(1)+1=4. See Figure 2.76(b).

Figure 2.76

A piecewise function

Combining these portions, we obtain the graph of y=F(x),y=F(x), shown in Figure 2.76(c). When we come to the end of the part of the graph we are working with, we draw

  1. a closed circle if that point is included.

  2. an open circle if the point is excluded.

You may find it helpful to think of this procedure as following the graph of y=−2x+1y=−2x+1 when x is less than 1 and then jumping to the graph of y=3x+1y=3x+1 when x is equal to or greater than 1.

Practice Problem 8

  1. Let f(x)={−3xif x≤−12xif x>−1.f(x)={−3x2xif x≤−1if x>−1. Sketch the graph of y=F(x).y=F(x).

Some piecewise functions are called step functions. Their graphs look like the steps of a staircase.

The greatest integer function is denoted by 〚x〛,〚x〛, or int(x), where 〚x〛=〚x〛= the greatest integer less than or equal to x. For example,

〚2〛=2,〚2.3〛=2,〚2.7〛=2,〚2.99〛=2
〚2〛=2,〚2.3〛=2,〚2.7〛=2,〚2.99〛=2

because 2 is the greatest integer less than or equal to 2, 2.3, 2.7, and 2.99. Similarly,

〚−2〛=−2,〚−1.9〛=−2,〚−1.1〛=−2,〚−1.001〛=−2
〚−2〛=−2,〚−1.9〛=−2,〚−1.1〛=−2,〚−1.001〛=−2

because −2−2 is the greatest integer less than or equal to −2,−1.9,−1.1,−2,−1.9,−1.1, and −1.001.−1.001.

In general, if m is an integer such that m≤x<m+1,m≤x<m+1, then 〚x〛=m.〚x〛=m. In other words, if x is between two consecutive integers m and m+1,m+1, then 〚x〛〚x〛 is assigned the smaller integer m.

Example 9 Graphing a Step Function

Graph the greatest integer function f(x)=〚x〛.f(x)=〚x〛.

Solution

Choose a typical closed interval between two consecutive integers—say, the interval [2, 3]. We know that between 2 and 3, the greatest integer function’s value is 2. In symbols, if 2≤x<3,2≤x<3, then 〚x〛=2.〚x〛=2. Similarly, if 1≤x<2,1≤x<2, then 〚x〛=1.〚x〛=1. Therefore, the values of 〚x〛〚x〛 are constant between each pair of consecutive integers and jump by one unit at each integer. The graph of f(x)=〚x〛f(x)=〚x〛 is shown in Figure 2.77.

Figure 2.77 Graph of y=〚x〛y=〚x〛

Practice Problem 9

  1. Find the values of f(x)=〚x〛f(x)=〚x〛 for x=−3.4x=−3.4 and x=4.7.x=4.7.

The greatest integer function f can be interpreted as a piecewise function:

f(x)=〚x〛={⋮−2if−2≤x<−1−1if−1≤x<00if0≤x<11if1≤x<2⋮
f(x)=〚x〛=⎧⎩⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪−2−101ifififif−2−101⋮≤≤≤≤⋮x<−1x<0x<1x<2

Basic Functions

  1. 4 Graph additional basic functions.

As you progress through this course and future mathematics courses, you will repeatedly come across a small list of basic functions. The following box lists some of these common algebraic functions, along with their properties. You should try to produce these graphs by plotting points and using symmetries. The unit length in all of the graphs shown is the same on both axes.

Section 2.6 Exercises

Concepts and Vocabulary

  1. The graph of the linear function f(x)=b is a(n)                            line.

  2. The absolute value function can be expressed as a piecewise function by writing f(x)=|x|=_.

  3. The graph of the function f(x)={x2+2if x≤1axif x>1 will have a break at x=1 unless a=_.

  4. The line that is the graph of f(x)=−2x+3 has slope                           .

  5. True or False. The graph of f(x)=mx+b cannot be a vertical line.

  6. True or False. No piecewise function has domain (−∞, ∞).

  7. True or False. The function f(x)=x2 is decreasing on (−∞, 0).

  8. True or False. The function f(x)=〚x〛 is increasing on the interval (0, 3).

Building Skills

In Exercises 9–18, write a linear function f that has the indicated values. Sketch the graph of f.

  1. f(0)=1, f(−1)=0

  2. f(1)=0, f(2)=1

  3. f(−1)=1, f(2)=7

  4. f(−1)=−5, f(2)=4

  5. f(1)=1, f(2)=−2

  6. f(1)=−1, f(3)=5

  7. f(−2)=2, f(2)=4

  8. f(2)=2, f(4)=5

  9. f(0)=−1, f(3)=−3

  10. f(1)=14, f(4)=−2

  11. Let

    f(x)={xif x≥22if x<2.

    1. Find f(1), f(2), and f(3).

    2. Sketch the graph of y=f(x).

  12. Let

    g(x)={2xif x<0xif x≥0.

    1. Find g(−1), g(0), and g(1).

    2. Sketch the graph of y=g(x).

  13. Let

    f(x)={1if x>0−1if x<0.

    1. Find f(−15) and f(12).

    2. Sketch the graph of y=f(x).

    3. Find the domain and the range of f.

  14. Let

    g(x)={2x+4if x>1x+4if x≤1.

    1. Find g(−3), g(1), and g(3).

    2. Sketch the graph of y=g(x).

    3. Find the domain and the range of g.

In Exercises 23–36, sketch the graph of each function and from the graph, find its domain and its range.

  1. f(x)={−2xif x<0x2if x≥0

  2. g(x)={|x|if x<1x2if x≥1

  3. g(x)={1xif x<0√xif x≥0

  4. h(x)={3√xif x<1√xif x≥1

  5. f(x)={〚x〛if x<13√xif x≥1

  6. g(x)={x3if x<0√xif x≥0

  7. g(x)={|x|if x<1x3if x≥1

  8. f(x)={1xif x<0〚x〛if x≥0

  9. f(x)={|x|if −2≤x<1√xif x≥1

  10. g(x)={3√xif −8≤x<12xif x≥1

  11. g(x)={〚x〛if x<1−2xif 1≤x≤4

  12. h(x)={|x|if x<0√xif 0≤x≤4

  13. f(x)={2x+3if x<−2x+1if −2≤x<1−x+3if x≥1

  14. g(x)={−2x+1if x≤−12x+1if −1<x<2x+2if x≥2

In Exercises 37–40, write a piecewise function for the given graph.

Applying the Concepts

  1. Converting volume. To convert the volume of a liquid measured in ounces to a volume measured in liters, we use the fact that 1 liter ≈33.81 ounces. Let x denote the volume measured in ounces and y denote the volume measured in liters.

    1. Write an equation for the linear function y=f(x). What are the domain and range of f?

    2. Compute f(3). What does it mean?

    3. How many liters of liquid are in a typical soda can containing 12 ounces of liquid?

  2. Boiling point and elevation. The boiling point B of water (in degrees Fahrenheit) at elevation h (in thousands of feet) above sea level is given by the linear function B(h)=−1.8h+212.

    1. Find and interpret the intercepts of the graph of the function y=B(h).

    2. Find the domain of this function.

    3. Find the elevation at which water boils at 98.6°F. Why is this height dangerous to humans?

  3. Pressure at sea depth. The pressure P in atmospheres (atm) at a depth d feet is given by the linear function

    P(d)=133d+1.

    1. Find and interpret the intercepts of y=P(d).

    2. Find P(0), P(10), P(33), and P(100).

    3. Find the depth at which the pressure is 5 atmospheres.

  4. Speed of sound. The speed V of sound (in feet per second) in air at temperature T (in degrees Fahrenheit) is given by the linear function V(T)=1055+1.1T.

    1. Find the speed of sound at 90°F.

    2. Find the temperature at which the speed of sound is 1100 feet per second.

  5. Manufacturer’s cost. A manufacturer of printers has a total cost per day consisting of a fixed overhead of $6000 plus product costs of $50 per printer.

    1. Express the total cost C as a function of the number x of printers produced.

    2. Draw the graph of y=C(x). Interpret the y-intercept.

    3. How many printers were manufactured on a day when the total cost was $11,500?

  6. Supply function. When the price p of a commodity is $10 per unit, 750 units of it are sold. For every $1 increase in the unit price, the supply q increases by 100 units.

    1. Write the linear function q=f(p).

    2. Find the supply when the price is $15 per unit.

    3. What is the price at which 1750 units can be supplied?

  7. Apartment rental. Suppose a two-bedroom apartment in a neighborhood near your school rents for $900 per month. If you move in after the first of the month, the rent is prorated; that is, it is reduced linearly.

    1. Suppose you move into the apartment x days after the first of the month. (Assume that the month has 30 days.) Express the rent R as a function of x.

    2. Compute the rent if you move in six days after the first of the month.

    3. When did you move into the apartment if the landlord charged you rent of $600?

  8. College admissions. The average SAT scores (mathematics and critical reading) of college-bound seniors in the state of Florida was 995 in 2009; it was 976 in 2011. Assume that the scores have been changing linearly.

    1. Express the average SAT score in Florida as a function of time since 2009.

    2. If the trend continued, what was the average SAT score of incoming students in Florida in 2013?

    3. If the trend continues, when will the average SAT score be 900?

  9. Breathing capacity. Suppose the average maximum breathing capacity for humans drops linearly from 100% at age 20 to 40% at age 80. What age corresponds to 50% capacity?

  10. Drug dosage for children. If a is the adult dosage of a medicine and t is the child’s age, then the Friend’s rule for the child’s dosage y is given by the formula y=225ta.

    1. Suppose the adult dosage is 60 milligrams. Find the dosage for a 5-year-old child.

    2. How old would a child have to be in order to be prescribed an adult dosage?

  11. Air pollution. In Ballerenia, the average number y of deaths per month was observed to be linearly related to the concentration x of sulfur dioxide in the air. Suppose there are 30 deaths when x=150 milligrams per cubic meter and 50 deaths when x=420 milligrams per cubic meter.

    1. Write y as a function of x.

    2. Find the number of deaths when x=350 milligrams per cubic meter.

    3. If the number of deaths per month is 70, what is the concentration of sulfur dioxide in the air?

  12. Child shoe sizes. Children’s shoe sizes start with size 0, having an insole length L of 31112 inches and each full size being 13 inch longer.

    1. Write the equation y=L(S), where S is the shoe size.

    2. Find the length of the insole of a child’s shoe of size 4.

    3. What size (to the nearest half size) shoe will fit a child whose insole length is 6.1 inches?

  13. State income tax. Suppose a state’s income tax code states that the tax liability T on x dollars of taxable income is as follows:

    T(x)={0.04xif 0≤x<20,000800+0.06(x−20,000)if x≥20,000

    1. Graph the function y=T(x).

    2. Find the tax liability on each taxable income.

      1. $12,000

      2. $20,000

      3. $50,000

    3. Find your taxable income if you had each tax liability.

      1. $600

      2. $1200

      3. $2300

  14. Federal income tax. The tax table for the 2015 U.S. income tax for a single taxpayer is as follows:

    If taxable income is over But not over The tax is
    $0 $9,225 10%
    9,225 37,450 $922.50+15% of the amount over— 9,225
    37,450 90,750 5,156.25+25% of the amount over— 37,450
    90,750 189,300 18,481.25+28% of the amount over— 90,750
    189,300 411,500 46,075.25+33% of the amount over— 189,300
    411,500 413,200 119,401.25+35% of the amount over— 411,500
    413,200 No Limit 119,996.25+39.6% of the amount over— 413,200

    Source: Internal Revenue Service.

    1. Express the information given in the table as a six-part piecewise-defined function f, using x as a variable representing taxable income.

    2. Find the tax liability on each taxable income. (Round your answer to the nearest dollar.)

      1. $35,000

      2. $100,000

      3. $500,000

    3. Find your taxable income if you had each tax liability. (Round your answer to the nearest dollar.)

      1. $3,500

      2. $12,700

      3. $35,000

Beyond the Basics

In Exercises 55 and 56, find the value of a such that the graph of the piecewise defined function f does not have any gaps (the two formulas that define f agree at the transition point).

  1. f(x)={2x−1if 1≤x≤3a−3xif 3<x≤5

  2. f(x)={1−xif 1≤x≤3ax+3if 3<x≤5

In Exercises 57 and 58, a function is given. For each function,

  1. find the domain and range.

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

  3. state whether the function is odd, even, or neither.

  1. f(x)=x−〚x〛

  2. f(x)=1〚x〛

  3. The windchill index (WCI). Suppose the outside air temperature is T degrees Fahrenheit and the wind speed is v miles per hour. A formula for the WCI based on observations is as follows:

    WCI={T,0≤v≤491.4+(91.4−T)(0.0203v−0.304√v −0.474),4<v<451.6T−55,v≥45

    1. Find the WCI to the nearest degree if the outside air temperature is 40°F and

      1. v=2 miles per hour.

      2. v=16 miles per hour.

      3. v=50 miles per hour.

    2. Find the air temperature to the nearest degree if

      1. the WCI is −58°F and v=36 miles per hour.

      2. the WCI is −10°F and v=49 miles per hour.

  4. Postage-rate function. The U.S. Postal Service uses the function f(x)=−〚−x〛, called the ceiling integer function, to determine postal charges. For instance, if you have an envelope that weighs 3.01 ounces, you will be charged the rate for f(3.01)=−〚−3.01〛=−(−4)=4 ounces. In 2012, it cost 45¢ for the first ounce (or a fraction of it) and 20¢ for each additional ounce (or a fraction of it) to mail a first-class letter in the United States.

    (Source: www.stamps.com/usps/postage-rate-increase)

    1. Express the cost C (in cents) of mailing a letter weighing x ounces.

    2. How much will it cost to mail a 2.3 oz first-class letter within the United States via USPS?

    3. Graph the function y=C(x) for 0≤x≤6.

  5. Parking cost. The cost C of parking a car at the metropolitan airport is $4 for the first hour and $2 for each additional hour or fraction thereof. Write the cost function C(x), where x is the number of parking hours, in terms of the greatest integer function.

  6. Car rentals. The weekly cost of renting a compact car from U-Rent is $150. There is no charge for driving the first 100 miles, but there is a 20¢ charge for each mile (or fraction thereof) driven over 100 miles.

    1. Express the cost C of renting a car for a week and driving it for x miles.

    2. Sketch the graph of y=C(x).

    3. How many miles were driven if the weekly rental cost was $190?

Critical Thinking / Discussion / Writing

In Exercises 63 and 64, select the graph that best describes the situation.

  1. An airplane flying from Tampa, Florida, to Atlanta, Georgia

  2. Cooling a jar of hot water by dumping a lot of ice

  3. Let the domain of a function f be (−∞, ∞) and let f be increasing on the interval (0, 1) and decreasing on the interval (1, ∞). What information can we deduce about the behavior of the function f on interval (−∞, 0) if

    1. we assume additionally that f is even?

    2. we assume additionally that f is odd?

  4. Let the domain of a function f be (−∞, ∞) and let f have a relative maximum at x=1 and relative minimum at x=3. What information can we deduce about the relative maxima and minima of the function f on the interval (−∞, 0) if

    1. we assume additionally that f is even?

    2. we assume additionally that f is odd?

Getting Ready for the Next Section

For Exercises 67–70, recall that the graph of a function f is the graph of the set of ordered pairs (x, f(x)).

  1. If we add 3 to each y-coordinate of the graph of f, we will obtain the graph of y=_.

  2. If we subtract 2 from each x-coordinate of the graph of f, we will obtain the graph of y=_.

  3. If we replace each x-coordinate with its opposite in the graph of f, we will obtain the graph of y=_.

  4. If we replace each y-coordinate with its opposite in the graph of f, we will obtain the graph of y=_.

In Exercises 71–74, sketch the graphs of f and g on the same coordinate axes.

  1. f(x)=x2 and g(x)=x2+1

  2. f(x)=|x| and g(x)=|x+2|

  3. f(x)=√x and g(x)=√−x

  4. f(x)=x2 and g(x)=−x2

In Exercises 75–78, write the number that must be added to the given expression so that the result is a perfect square trinomial. Write the result in the form (x+b)2.

  1. x2+8x

  2. x2−6x

  3. x2−23x

  4. x2+45x

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