1.2 Functions and Graphs

  • Determine whether a correspondence or a relation is a function.

  • Find function values, or outputs, using a formula or a graph.

  • Graph functions.

  • Determine whether a graph is that of a function.

  • Find the domain and the range of a function.

  • Solve applied problems using functions.

We now focus our attention on a concept that is fundamental to many areas of mathematics—the idea of a function.

Functions

Used-Book Co-op. A community center operates a used-book co-op, and the proceeds are donated to summer youth programs. The total cost of a purchase is $2.50 per book plus a flat-rate surcharge of $3. If a customer selects 6 books, the total cost of the purchase is

$2.50(6)+$3, or $18.

We can express this relationship with a set of ordered pairs, a graph, and an equation. A few ordered pairs are listed in the following table.

x y Ordered Pairs: (x, y) Correspondence
1 5.50 (1, 5.50) 1→5.50
2 8.00 (2, 8) 2→8
4 13.00 (4, 13) 4→13
7 20.50 (7, 20.50) 7→20.50
10 28.00 (10, 28) 10→28

The ordered pairs express a relationship, or a correspondence, between the first coordinate and the second coordinate. We can see this relationship in the graph as well. The equation that describes the correspondence is

y=2.50x+3, where x is a natural number.

This is an example of a function. In this case, the total cost of the purchase y is a function of the number of books purchased x; that is, y is a function of x, where x is the independent variable and y is the dependent variable.

Let’s consider some other correspondences before giving the definition of a function.

First Set Correspondence Second Set
To each person there corresponds that person’s DNA.
To each blue spruce sold there corresponds its price.
To each real number there corresponds the square of that number.

In each correspondence, the first set is called the domain and the second set is called the range. For each member, or element, in the domain, there is exactly one member in the range to which it corresponds. Thus each person has exactly one DNA, each blue spruce has exactly one price, and each real number has exactly one square. Each correspondence is a function.

It is important to note that not every correspondence between two sets is a function.

Example 1

Determine whether each of the following correspondences is a function.

Solution

  1. This correspondence is a function because each member of the domain corresponds to exactly one member of the range. Note that the definition of a function allows more than one member of the domain to correspond to the same member of the range.

  2. This correspondence is not a function because there is at least one member of the domain who is paired with more than one member of the range (William Jefferson Clinton with Stephen G. Breyer and Ruth Bader Ginsburg; George W. Bush with Samuel A. Alito, Jr., and John G. Roberts, Jr.; Barack H. Obama with Elena Kagan and Sonia M. Sotomayor).

Now Try Exercises 5 and 7.

Example 2

Determine whether each of the following correspondences is a function.

DOMAIN CORRESPONDENCE RANGE
a) Years in which a presidential election occurs The person elected A set of presidents
b) All automobiles produced in 2014 Each automobile’s VIN (Vehicle Identification Number) A set of VINs
c) The set of all professional golfers who won a PGA tournament in 2013 The tournament won The set of all PGA tournaments in 2013
d) The set of all PGA tournaments in 2013 The winner of the tournament The set of all golfers who won a PGA tournament in 2013

Solution

  1. This correspondence is a function because in each presidential election exactly one president is elected.

  2. This correspondence is a function because each automobile has exactly one VIN.

  3. This correspondence is not a function because a winning golfer could be paired with more than one tournament.

  4. This correspondence is a function because each tournament has only one winning golfer.

Now Try Exercises 11 and 13.

When a correspondence between two sets is not a function, it may still be an example of a relation.

All the correspondences in Examples 1 and 2 are relations, but, as we have seen, not all are functions. Relations are sometimes written as sets of ordered pairs (as we saw earlier in the example on the total cost of a purchase of used books) in which elements of the domain are the first coordinates of the ordered pairs and elements of the range are the second coordinates. For example, instead of writing −3→9, as we did in Example 1(a), we could write the ordered pair (−3, 9).

Example 3

Determine whether each of the following relations is a function. Identify the domain and the range.

  1. {(9, −5), (9, 5), (2, 4)}

  2. {(−2, 5), (5, 7), (0, 1), (4, −2)}

  3. {(−5, 3), (0, 3), (6, 3)}

Solution

  1. The relation is not a function because the ordered pairs (9, −5) and (9, 5) have the same first coordinate and different second coordinates. (See Fig. 1.)

    Figure 1.

    • The domain is the set of all first coordinates: {9, 2}.

    • The range is the set of all second coordinates: {−5, 5, 4}.

  2. The relation is a function because no two ordered pairs have the same first coordinate and different second coordinates. (See Fig. 2.)

    Figure 2.

    • The domain is the set of all first coordinates: {−2, 5, 0, 4}.

    • The range is the set of all second coordinates: {5, 7, 1, −2}.

  3. The relation is a function because no two ordered pairs have the same first coordinate and different second coordinates. (See Fig. 3.)

    Figure 3.

    • The domain is {−5, 0, 6}.

    • The range is {3}.

Now Try Exercises 15 and 17.

Notation for Functions

Functions used in mathematics are often given by equations. They generally require that certain calculations be performed in order to determine which member of the range is paired with each member of the domain. For example, in Section 1.1 we graphed the function y=x2−9x−12 by doing calculations like the following:

  • for x=−2, y=(−2)2−9(−2)−12=10,

  • for x=0, y=02−9⋅0−12=−12, and

  • for x=1, y=12−9⋅1−12=−20.

A more concise notation is often used. For y=x2−9x−12, the inputs (members of the domain) are values of x substituted into the equation. The outputs (members of the range) are the resulting values of y. If we call the function f, we can use x to represent an arbitrary input and f(x)—read “f of x,” or “f at x,” or “the value of f at x”—to represent the corresponding output. In this notation, the function given by y=x2−9x−12 is written as f(x)=x2−9x−12 and the above calculations would be

  • f(−2)=(−2)2 − 9(−2) − 12=10,

  • f(0)=02 − 9⋅0 − 12=−12, and

  • f(1)=12 − 9⋅1 − 12=−20.Keep in mind that f(x)does not mean f⋅x.

Thus, instead of writing “when x=−2, the value of y is 10,” we can simply write “f(−2)=10,” which can be read as “f of −2 is 10” or “for the input −2, the output of f is 10.” The letters g and h are also often used to name functions.

Example 4

A function f is given by f(x)=2x2−x+3. Find each of the following.

  1. f(0)

  2. f(−7)

  3. f(5a)

  4. f(a−4)

Solution

We can think of this formula as follows:

f()=2()2−()+3.

Then to find an output for a given input, we think: “Whatever goes in the blank on the left goes in the blank(s) on the right.” This gives us a “recipe” for finding outputs.

  1. f(0)=2(0)2−0+3=0−0+3=3

  2. f(−7)=2(−7)2−(−7)+3=2⋅49+7+3=108

  3. f(5a)=2(5a)2−5a+3=2⋅25a2−5a+3=50a2−5a+3

  4. f(a−4)=2(a−4)2−(a−4)+3=2(a2−8a+16)−(a−4)+3=2a2−16a+32−a+4+3=2a2−17a+39

Now Try Exercise 21.

Graphs of Functions

We graph functions in the same way that we graph equations. We find ordered pairs (x, y), or (x, f(x)), plot points, and complete the graph.

Example 5

Graph each of the following functions.

  1. f(x)=x2−5

  2. f(x)=x3−x

  3. f(x)=√x+4

Solution

We select values for x and find the corresponding values of f(x). Then we plot the points and connect them with a smooth curve.

  1. f(x)=x2−5

    x f(x) (x, f(x))
    −3 4 (−3, 4)
    −2 −1 (−2, −1)
    −1 −4 (−1, −4)
    0 −5 (0, −5)
    1 −4 (1, −4)
    2 −1 (2, −1)
    3 4 (3, 4)

  2. f(x)=x3−x

  3. f(x)=√x+4

Now Try Exercise 31.

Function values can also be determined from a graph.

Example 6

For the function f(x)=x2−6, use the graph at left to find each of the following function values.

  1. f(−3)

  2. f(1)

Solution

  1. To find the function value f(−3) from the graph, we locate the input −3 on the horizontal axis, move vertically to the graph of the function, and then move horizontally to find the output on the vertical axis. We see that f(−3)=3.

  2. To find the function value f(1), we locate the input 1 on the horizontal axis, move vertically to the graph, and then move horizontally to find the output on the vertical axis. We see that f(1)=−5.

Now Try Exercise 35.

We know that when one member of the domain is paired with two or more different members of the range, the correspondence is not a function. Thus, when a graph contains two or more different points with the same first coordinate, the graph cannot represent a function. (See the graph at left. Note that 3 is paired with −1, 2, and 5.) Points sharing a common first coordinate are vertically above or below each other. This leads us to the vertical-line test.

Since 3 is paired with more than one member of the range, the graph does not represent a function.

To apply the vertical-line test, we try to find a vertical line that crosses the graph more than once. If we succeed, then the graph is not that of a function. If we do not, then the graph is that of a function.

Example 7

Which of graphs (a)–(f) (in red) are graphs of functions? In graph (f), the solid dot shows that (−1, 1) belongs to the graph. The open circle shows that (−1, −2) does not belong to the graph.

Solution

Graphs (a), (e), and (f) are graphs of functions because we cannot find a vertical line that crosses any of them more than once. In (b), the vertical line drawn crosses the graph at three points, so graph (b) is not that of a function. Also, in (c) and (d), we can find a vertical line that crosses the graph more than once, so these are not graphs of functions.

Now Try Exercises 43 and 47.

Finding Domains of Functions

When a function f whose inputs and outputs are real numbers is given by a formula, the domain is understood to be the set of all inputs for which the expression is defined as a real number. When an input results in an expression that is not defined as a real number, we say that the function value does not exist and that the number being substituted is not in the domain of the function.

Example 8

Find the indicated function values, if possible, and determine whether the given values are in the domain of the function.

  1. f(1) and f(3), for f(x)=1x−3

  2. g(16) and g(−7), for g(x)=√x+5

Solution

  1. f(1)=11−3=1−2=−12

    Since f(1) is defined, 1 is in the domain of f.

    f(3)=13−3=10

    Since division by 0 is not defined, f(3) does not exist and the number 3 is not in the domain of f.

  2. g(16)=√16+5=4+5=9

    Since g(16) is defined, 16 is in the domain of g.

    g(−7)=√−7+5

    Since √−7 is not defined as a real number, g(−7) does not exist and the number −7 is not in the domain of g.

As we see in Example 8, inputs that make a denominator 0 or that yield a negative radicand in an even root are not in the domain of a function.

Example 9

Find the domain of each of the following functions.

  1. f(x)=1x−7

  2. h(x)=3x2−x+7x2+2x−3

  3. f(x)=x3+|x|

  4. g(x)=3√x−1

Solution

  1. Because x−7=0 when x=7, the only input that results in a denominator of 0 is 7. The domain is {x|x≠7}. We can also write the solution using interval notation and the symbol ∪ for the union, or inclusion, of both sets: (−∞, 7) ∪ (7, ∞).

  2. We can substitute any real number in the numerator, but we must avoid inputs that make the denominator 0. To find those inputs, we solve x2+2x−3=0, or (x+3)(x−1)=0. Since x2+2x−3 is 0 for −3 and 1, the domain consists of the set of all real numbers except −3 and 1, or {x|x≠−3 and x≠1}, or (−∞, −3) ∪ (−3, 1)∪(1, ∞).

  3. We can substitute any real number for x. Thus the domain is the set of all real numbers, ℝ, or (−∞, ∞).

  4. Because the index is odd, the radicand, x−1, can be any real number. Thus x can be any real number. The domain is all real numbers, ℝ, or (−∞, ∞).

Now Try Exercises 55, 57, and 61.

Visualizing Domain and Range

Consider the graph of function f, shown at left. To determine the domain of f, we look for the inputs on the x-axis that correspond to a point on the graph. We see that they include the entire set of real numbers, illustrated in red on the x-axis. Thus the domain is (−∞, ∞). To find the range, we look for the outputs on the y-axis that correspond to a point on the graph. We see that they include 4 and all real numbers less than 4, illustrated in blue on the y-axis. The bracket at 4 indicates that 4 is included in the interval. The range is {y|y≤4}, or (−∞, 4].

Let’s now consider the following graph of function g. The solid dot shows that (−4, 5) belongs to the graph. The open circle shows that (3, 2) does not belong to the graph.

We see that the inputs of the function include −4 and all real numbers between −4 and 3, illustrated in red on the x-axis. The bracket at −4 indicates that −4 is included in the interval. The parenthesis at 3 indicates that 3 is not included in the interval. The domain is {x|−4≤x<3}, or [−4, 3). The outputs of the function include 5 and all real numbers between 2 and 5, illustrated in blue on the y-axis. The parenthesis at 2 indicates that 2 is not included in the interval. The bracket at 5 indicates that 5 is included in the interval. The range is {y|2<y≤5}, or (2, 5].

Example 10

Using the graph of the function, find the domain and the range of the function.

  1. f(x)=12x+1

  2. f(x)=√x+4

  3. f(x)=x3−x

  4. f(x)=1x−2

  5. f(x)=x4−2x2−3

  6. f(x)=√4−(x−3)2

Solution

  1. Domain = all real numbers, (−∞, ∞); range = all real numbers, (−∞, ∞)

  2. Domain=[−4, ∞); range=[0, ∞)

  3. Domain = all real numbers, (−∞, ∞); range=all real numbers, (−∞, ∞)

  4. Since the graph does not touch or cross either the vertical line x=2 or the x-axis, y=0, 2 is excluded from the domain and 0 is excluded from the range. Domain=(−∞, 2)∪(2, ∞); range=(−∞, 0)∪(0, ∞)

  5. Domain = all real numbers, (−∞, ∞); range=[−4, ∞)

  6. Domain=[1, 5]; range=[0, 2]

Now Try Exercises 71 and 77.

Always consider adding the reasoning of Example 9 to a graphical analysis. Think, “What can I input?” to find the domain. Think, “What do I get out?” to find the range. Thus, in Examples 10(c) and 10(e), it might not appear as though the domain is all real numbers because the graph rises steeply, but by examining the equation we see that we can indeed substitute any real number for x.

Applications of Functions

Example 11

Linear Expansion of a Bridge. The linear expansion L of the steel center span of a suspension bridge that is 1420 m long is a function of the change in temperature t, in degrees Celsius, from winter to summer and is given by

L(t)=0.000013⋅1420⋅t,

where 0.000013 is the coefficient of linear expansion for steel and L is in meters. Find the linear expansion of the steel center span when the change in temperature from winter to summer is 30°, 42°, 50°, and 56° Celsius.

Solution

Using a calculator, we compute function values. We find that

L(30)=0.5538 m,L(42)=0.77532 m,L(50)=0.923 m, andL(56)=1.03376 m.

Now Try Exercise 85.

1.2 Exercise Set

In Exercises 1–14, determine whether the correspondence is a function.

  1. 1.

  2. 2.

  3. 3.

  4. 4.

  5. 5.

  6. 6.

  7. 7.

  8. 8.

  9.   DOMAIN CORRESPONDENCE RANGE
    9.

    A set of cars in a parking lot

    Each car’s license number

    A set of letters and numbers
    10.

    A set of people in a town

    A doctor a person uses

    A set of doctors

    11.

    		 The integers less than 9

    Five times the integer

    A subset of integers
    12.

    A set of members of a rock band

    An instrument each person plays

    A set of instruments

    13.

    A set of students in a class

    A student sitting in a neighboring seat

    A set of students
    14.

    A set of bags of chips on a shelf

    Each bag’s weight

    A set of weights

Determine whether the relation is a function. Identify the domain and the range.

  1. 15. {(2, 10), (3, 15), (4, 20)}

  2. 16. {(3, 1), (5, 1), (7, 1)}

  3. 17. {(−7, 3), (−2, 1), (−2, 4), (0, 7)}

  4. 18. {(1, 3), (1, 5), (1, 7), (1, 9)}

  5. 19. {(−2, 1), (0, 1), (2, 1), (4, 1), (−3, 1)}

  6. 20. {(5, 0), (3, −1), (0, 0), (5, −1), (3, −2)}

  7. 21. Given that g(x)=3x2−2x+1, find each of the following.

    1. g(0)

    2. g(−1)

    3. g(3)

    4. g(−x)

    5. g(1−t)

  8. 22. Given that f(x)=5x2+4x, find each of the following.

    1. f(0)

    2. f(−1)

    3. f(3)

    4. f(t)

    5. f(t−1)

  9. 23. Given that g(x)=x3, find each of the following.

    1. g(2)

    2. g(−2)

    3. g(−x)

    4. g(3y)

    5. g(2+h)

  10. 24. Given that f(x)=2|x|+3x, find each of the following.

    1. f(1)

    2. f(−2)

    3. f(−x)

    4. f(2y)

    5. f(2−h)

  11. 25. Given that

    g(x)=x−4x+3,

    find each of the following.

    1. g(5)

    2. g(4)

    3. g(−3)

    4. g(−16.25)

    5. g(x+h)

  12. 26. Given that

    f(x)=x2−x,

    find each of the following.

    1. f(2)

    2. f(1)

    3. f(−16)

    4. f(−x)

    5. f(−23)

  13. 27. Find g(0), g(−1), g(5), and g(12) for

    g(x)=x√1−x2.

  14. 28. Find h(0), h(2), and h(−x) for

    h(x)=x+√x2−1.

Graph the function.

  1. 29. f(x)=12x+3

  2. 30. f(x)=√x−1

  3. 31. f(x)=−x2+4

  4. 32. f(x)=x2+1

  5. 33. f(x)=√x−1

  6. 34. f(x)=x−12x3

In each of Exercises 35–40, a graph of a function is shown. Using the graph, find the indicated function values; that is, given the inputs, find the outputs.

  1. 35. h(1), h(3), and h(4)

  2. 36. t(−4), t(0), and t(3)

  3. 37. s(−4), s(−2), and s(0)

  4. 38. g(−4), g(−1), and g(0)

  5. 39. f(−1), f(0), and f(1)

  6. 40. g(−2), g(0), and g(2.4)

In Exercises 41–48, determine whether the graph is that of a function. An open circle indicates that the point does not belong to the graph.

  1. 41.

  2. 42.

  3. 43.

  4. 44.

  5. 45.

  6. 46.

  7. 47.

  8. 48.

Find the domain of the function.

  1. 49. f(x)=7x+4

  2. 50. f(x)=|3x−2|

  3. 51. f(x)=|6−x|

  4. 52. f(x)=1x4

  5. 53. f(x)=4−2x

  6. 54. f(x)=15x2−5

  7. 55. f(x)=x+52−x

  8. 56. f(x)=8x+4

  9. 57. f(x)=1x2−4x−5

  10. 58. f(x)=(x−2)(x+9)x3

  11. 59. f(x)=3√x+10−1

  12. 60. f(x)=3√4−x

  13. 61. f(x)=8−xx2−7x

  14. 62. f(x)=x4−2x3+73x2−10x−8

  15. 63. f(x)=110|x|

  16. 64. f(x)=x2−2x

In Exercises 65–72, determine the domain and the range of the function.

  1. 65.

  2. 66.

  3. 67.

  4. 68.

  5. 69.

  6. 70.

  7. 71.

  8. 72.

In Exercises 73–84, graph the given function. Then visually estimate the domain and the range.

  1. 73. f(x)=|x|

  2. 74. f(x)=|x|−2

  3. 75. f(x)=3x−2

  4. 76. f(x)=5−3x

  5. 77. f(x)=1x−3

  6. 78. f(x)=1x+1

  7. 79. f(x)=(x−1)3+2

  8. 80. f(x)=(x−2)4+1

  9. 81. f(x)=√7−x

  10. 82. f(x)=√x+8

  11. 83. f(x)=−x2+4x−1

  12. 84. f(x)=2x2−x4+5

  13. 85. Decreasing Value of the Dollar. In 2014, it took $23.63 to equal the value of $1 in 1913. In 2000, it took only $17.39 to equal the value of $1 in 1913. The amount that it takes to equal the value of $1 in 1913 can be estimated by the linear function V given by

    V(x)=0.4306x+11.0043,

    where x is the number of years since 1985. Thus, V(10) gives the amount that it took in 1995 to equal the value of $1 in 1913.

    Source: usinflationcalculator.com

    1. Use this function to predict the amount that it will take in 2018 and in 2025 to equal the value of $1 in 1913.

    2. When will it take approximately $32 to equal the value of $1 in 1913?

  14. 86. Population of the United States. The population P of the United States in 1960 was 179,323,175. In 2010, the population was 308,745,538. The population of the United States can be estimated by the linear function P given by

    P(x)=2,578,409x+151,116,864,

    where x is the number of years after 1950. Thus, P(20) gives the population in 1970.

    1. Use this function to estimate the population in 1980 and in 2018.

    2. When will the population be approximately 400,000,000?

  15. 87. Boiling Point and Elevation. The elevation E, in meters, above sea level at which the boiling point of water is t degrees Celsius is given by the function

    E(t)=1000(100−t)+580(100−t)2.

    At what elevation is the boiling point 99.5°? 100°?

  16. 88. Windmill Power. Under certain conditions, the power P, in watts per hour, generated by a windmill with winds blowing v miles per hour is given by

    P(v)=0.015v3.

    Find the power generated by 15-mph winds and 35-mph winds.

Skill Maintenance

To the student and the instructor: The Skill Maintenance exercises review skills covered previously in the text. You can expect such exercises in every exercise set. They provide excellent review for a final examination. Answers to all skill maintenance exercises, along with section references, appear in the answer section at the back of the book.

Use substitution to determine whether the given ordered pairs are solutions of the given equation.

  1. 89. (−3, −2), (2, −3); y2−x2=−5 [1.1]

  2. 90. (0, −7), (8, 11); y=0.5x+7 [1.1]

  3. 91. (45, −2), (115, 1 10); 15x−10y=32 [1.1]

Graph the equation. [1.1]

  1. 92. y=(x−1)2

  2. 93. y=13x−6

  3. 94. −2x−5y=10

  4. 95. (x−3)2+y2=4

Synthesis

Find the domain of the function.

  1. 96. f(x)=4√2x+5+3

  2. 97. f(x)=√x+1x

  3. 98. f(x)=√x+6(x+2)(x−3)

  4. 99. f(x)=√x−√4−x

  5. 100. Give an example of two different functions that have the same domain and the same range, but have no pairs in common. Answers may vary.

  6. 101. Draw a graph of a function for which the domain is [−4, 4] and the range is [1, 2]∪[3, 5]. Answers may vary.

  7. 102. Suppose that for some function g, g(x+3)=2x+1. Find g(−1).

  8. 103. Suppose f(x)=|x+3|−|x−4|. Write f(x) without using absolute-value notation if x is in each of the following intervals.

    1. (−∞, −3)

    2. [−3, 4)

    3. [4, ∞)

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