Determine whether each relation is a function. Explain.
1.
SOLUTION:
A function is a relation in which each element of the domain is paired with exactly one element of the range. So, this
relation is a function.
ANSWER:
Yes; for each input there is exactly one output.
2.
SOLUTION:
A function is a relation in which each element of the domain is paired with exactly one element of the range. In the
domain, the value 6 is paired with both 9 and 10. So, this relation is not a function.
ANSWER:
No; the domain value 6 is paired with both 9 and 10.
3.{(2, 2), (−1, 5), (5, 2), (2, −4)}
SOLUTION:
A function is a relation in which each element of the domain is paired with exactly one element of the range. In the
domain, the value 2 is paired with 2 and −4. So, this relation is not a function.
ANSWER:
No; the domain value 2 is paired with 2 and −4.
4.
SOLUTION:
This is a function because no vertical line can be drawn so that it intersects the graph more than once.
ANSWER:
Yes; the graph passes the vertical line test.
5.
SOLUTION:
A function is a relation in which each element of the domain is paired with exactly one element of the range. When x
= 0, y = 1 and y = 6. So, this relation is not a function.
ANSWER:
No; when x = 0, y = 1 and y = 6.
6.
SOLUTION:
This is a function because no vertical line can be drawn so that it intersects the graph more than once.
ANSWER:
Yes; the graph passes the vertical line test.
7.
SOLUTION:
This is a function because no vertical line can be drawn so that it intersects the graph more than once.
ANSWER:
Yes; the graph passes the vertical line test.
8.
SOLUTION:
This is not a function because a vertical line can be drawn so that it intersects the graph more than once.
ANSWER:
No; the graph does not pass the vertical line test.
9.SCHOOL ENROLLMENT The table shows the total enrollment in U.S. public schools.
a. Write a set of ordered pairs representing the data in the table if x is the number of school years since 2004-2005.
b. Draw a graph showing the relationship between the year and enrollment.
c. Describe the domain and range of the data.
SOLUTION:
a. The school year is the domain for this relation. The enrollment is the range. So, when creating ordered pairs, the
school year is first and the enrollment is second. The ordered pairs for this data are {(0, 48,560), (1, 48,710), (2,
48,948), (3, 49,091)}.
b.
c. The domain is the school year and the range is the enrollment.
ANSWER:
a. {(0, 48,560), (1, 48,710), (2, 48,948), (3, 49,091)}
b.
c. The domain is the school year and the range is the enrollment.
10.CCSS REASONING The cost of sending cell phone pictures is given by y = 0.25x, where x is the number of
pictures sent, and y isthecostindollars.
a. Writetheequationinfunctionnotation.Interpretthefunctionintermsofthecontext.
b. Find f(5) and f(12).Whatdothesevaluesrepresent?
c. Determine the domain and range of this function.
SOLUTION:
In function notation, f(x) represents the range. So, the function looks like f(x) = 0.25x written in function notation.
So, it costs $1.25 to send 5 photos and $3.00 to send 12 photos. The domain is the number of pictures sent and the
cost is the range.
ANSWER:
a.f(x) = 0.25x
b. f(5) = $1.25, f(12) = $3.00; These values represent the cost to send five and twelve pictures, respectively.
c. The domain is the number of pictures sent and the cost is the range.
If f(x) = 6x + 7 and g(x) = x2 − 4, find each value.
11.f(−3)
SOLUTION:
ANSWER:
−11
12.f(m)
SOLUTION:
ANSWER:
6m + 7
13.f(r − 2)
SOLUTION:
ANSWER:
6r − 5
14.g(5)
SOLUTION:
ANSWER:
21
15.g(a) + 9
SOLUTION:
ANSWER:
a2 + 5
16.g(−4t)
SOLUTION:
ANSWER:
16t2 −4
17.f(q + 1)
SOLUTION:
ANSWER:
6q + 13
18.f(2) + g(2)
SOLUTION:
ANSWER:
19
19.g(−b)
SOLUTION:
ANSWER:
b2 − 4
Determine whether each relation is a function. Explain.
20.
SOLUTION:
A function is a relation in which each element of the domain is paired with exactly one element of the range. So, this
relation is a function.
ANSWER:
yes; for each input there is exactly one output.
21.
SOLUTION:
A function is a relation in which each element of the domain is paired with exactly one element of the range. In the
domain, the value 4 is paired with both 5 and 6. So, this relation is not a function.
ANSWER:
no; the domain value 4 is paired with both 5 and 6.
22.
SOLUTION:
A function is a relation in which each element of the domain is paired with exactly one element of the range. In the
domain, the value −5 is paired with both 3 and 5. So, this relation is not a function.
ANSWER:
no; the domain value −5 is paired with both 3 and 5.
23.
SOLUTION:
A function is a relation in which each element of the domain is paired with exactly one element of the range. So, this
relation is a function.
ANSWER:
yes; for each input there is exactly one output.
24.
SOLUTION:
A function is a relation in which each element of the domain is paired with exactly one element of the range. When x
= 4, y = 4 and y = 6. So, this relation is not a function.
ANSWER:
no; when x = 4, y = 4 and y = 6.
25.
SOLUTION:
This is a function because no vertical line can be drawn so that it intersects the graph more than once.
ANSWER:
yes; the graph passes the vertical line test.
26.CCSS SENSE-MAKING The table shows the median home prices in the United States, from 2007 to 2009.
a. Write a set of ordered pairs representing the data in the table.
b. Draw a graph showing the relationship between the year and price.
c. What is the domain and range for this data?
SOLUTION:
a. The year is the domain for this relation. The median price is the range. So, when creating ordered pairs, the year is
first and the median price is second. The ordered pairs for this data are {(2007, 234,300), (2008, 213,200), (2009,
212,200)}.
b.
c. The domain is the year. The range is the median home price.
ANSWER:
a. {(2007, 234,300), (2008, 213,200), (2009, 212,200)}
b.
c. The domain is the year. The range is the median home price.
Determine whether each relation is a function.
27.{(5, −7), (6, −7), (−8, −1), (0, −1)}
SOLUTION:
A function is a relation in which each element of the domain is paired with exactly one element of the range. So, this
relation is a function.
ANSWER:
yes
28.{(4, 5), (3, −2), (−2, 5), (4, 7)}
SOLUTION:
A function is a relation in which each element of the domain is paired with exactly one element of the range. The
value 4 is paired with 5 and 7. So, this relation is not a function.
ANSWER:
no
29.y = −8
SOLUTION:
This is a function because no vertical line can be drawn so that it intersects the graph more than once.
ANSWER:
yes
30.x = 15
SOLUTION:
This is not a function because a vertical line can be drawn so that it intersects the graph more than once.
ANSWER:
no
31.y = 3x −2
SOLUTION:
This is a function because no vertical line can be drawn so that it intersects the graph more than once.
ANSWER:
yes
32.y = 3x + 2y
SOLUTION:
This is a function because no vertical line can be drawn so that it intersects the graph more than once.
ANSWER:
yes
If f(x) = −2x − 3 and g(x) = x2 + 5x, find each value.
33.f(−1)
SOLUTION:
ANSWER:
−1
34.f(6)
SOLUTION:
ANSWER:
−15
35.g(2)
SOLUTION:
ANSWER:
14
36.g(−3)
SOLUTION:
ANSWER:
−6
37.g(−2) + 2
SOLUTION:
ANSWER:
−4
38.f(0) − 7
SOLUTION:
ANSWER:
−10
39.f(4y)
SOLUTION:
ANSWER:
−8y − 3
40.g(−6m)
SOLUTION:
ANSWER:
36m2 − 30m
41.f(c − 5)
SOLUTION:
ANSWER:
−2c + 7
42.f(r + 2)
SOLUTION:
ANSWER:
−2r − 7
43.5[f(d)]
SOLUTION:
ANSWER:
−10d − 15
44.3[g(n)]
SOLUTION:
ANSWER:
3n2 + 15n
45.EDUCATION The average national math test scores f(t) for 17-year-olds can be represented as a function of the
national science scores t by f(t) = 0.8t + 72.
a. Graph this function. Interpret the function in terms of the context.
b. What is the science score that corresponds to a math score of 308?
c. What is the domain and range of this function?
SOLUTION:
a.
Whenthesciencescoreis0,themathscoreis72.Foreachpointthesciencescoreincreases,
the math score increases by 0.8 point.
b.
c. The domain is the independent variable or x-variable. Thus the domain is the set of science scores. The range is
the dependent variable or the y-variable. Thus, the range is the set of math scores.
ANSWER:
a.
Whenthesciencescoreis0,themathscoreis72.Foreachpointthesciencescoreincreases,
the math score increases by 0.8 point.
b. 295
c. The domain is the set of science scores. The range is the set of math scores.
Determine whether each relation is a function.
46.
SOLUTION:
This is a function because no vertical line can be drawn so that it intersects the graph more than once.
ANSWER:
yes
47.
SOLUTION:
This is a function because no vertical line can be drawn so that it intersects the graph more than once.
ANSWER:
yes
48.BABYSITTING Christina earns $7.50 an hour babysitting.
a. Write an algebraic expression to represent the money Christina will earn if she works h hours.
b. Choose five values for the number of hours Christina can babysit. Create a table with h and the value for the
amount of money she will make during that time.
c. Use the values in your table to create a graph.
d. Does it make sense to connect the points in your graph with a line? Why or why not?
SOLUTION:
a. To find the money Christina will make babysitting, multiply the number of hours she worked by her pay or $7.50.
So, an algebraic expression to represent the money she earns is $7.50h.
b. Sample answer:
c.
d. By connecting the points, all values between the numbers are included. So, the points should be connected
because they could pay Christina for partial hours that she worked.
ANSWER:
a. $7.50h
b. Sample answer:
c.
d. Yes, because they could pay Christina for partial hours that she worked.
49.OPEN ENDED Write a set of three ordered pairs that represent a function. Choose another display that represents
this function.
SOLUTION:
{(–2, 3), (0, 3), (2, 5)} is a set of ordered pairs.
A mapping is another display that represents the function. Place the x-coordinates in the domain and the y-
coordinates in the range. Link each value in the domain with the corresponding value in the range.
ANSWER:
Sample answer: {(–2, 3), (0, 3), (2, 5)};
50.REASONING The set of ordered pairs {(0, 1), (3, 2), (3, −5), (5, 4)} represents a relation between x and y. Graph
the set of ordered pairs. Determine whether the relation is a function. Explain.
SOLUTION:
A function is a relation in which each element of the domain is paired with exactly one element of the range. The
value 3 is paired with −5 and 2. So, this relation is not a function.
ANSWER:
Not a function; one member of the domain, 3, is paired with two different members of the range, −5 and 2.
51.CHALLENGE Consider f(x) = −4.3x − 2. Write f(g + 3.5) and simplify by combining like terms.
SOLUTION:
ANSWER:
f(g + 3.5) = −4.3g − 17.05
52.WRITE A QUESTION A classmate graphed a set of ordered pairs and used the vertical line test to determine
whether it was a function. Write a question to help her decide if the same strategy can be applied to a mapping.
SOLUTION:
If the classmate could see the ordered pair in another way, it could help her see the relation. So, "Isn’t a mapping
another representation of a set of ordered pairs?“is a good question to ask.
ANSWER:
Sample answer: Isn’t a mapping another representation of a set of ordered pairs?
53.CCSS PERSEVERENCE If f(3b − 1) = 9b − 1, find one possible expression for f(x).
SOLUTION:
Our input is “3b – 1”and our output is “9b – 1”. We need to find a function that convert the input to the output.
Let’s look at the variable first. What can we do to 3b to change it to 9b?3×3=9,solet’s multiply by 3. Let f(x) =
3x. Test the function.
We did not get 9b – 1, but we were close. How can we change the “–3”to a “–1”? Add 2. Now, let’s try f(x) = 3x
+ 2.
We have found a function that works.
ANSWER:
Sample answer: f(x) = 3x + 2
54.ERROR ANALYSIS Corazon thinks f(x) and g(x) are representations of the same function. Maggie disagrees.
Who is correct? Explain your reasoning.
SOLUTION:
The graph has a y-intercept of 1. It also contains the point (1, –1), which we can use to determine the slope:
The equation for f(x) is: f(x) = –2x+1.
For the table, we can see that as x increases by 1, g(x) decreases by 2, which means the slope of g(x) is –2. But the
y-intercept for g(x) is (0, –1), giving g(x) = –2x – 1.
Thegraphandtablearerepresentativeofdifferentfunctions.
ANSWER:
Maggie; the graph represents f (x ) = –2x + 1 and the table represents g(x ) = –2x – 1.
55.WRITING IN MATH How can you determine whether a relation represents a function?
SOLUTION:
A relation is a function if each element of the domain is paired with exactly one element of the range. If given a
graph,thismeansthatitmustpasstheverticallinetest.
Function
Not a function
If given a table, or a set of ordered pairs, you can look to see if any value of the domain has more than one
correspondingvalueintherange.
Function
Not a function
x y
1–3
2–6
3–9
4–12
x y
–1 2
0 3
0–3
1–2
ANSWER:
Sample answer: You can determine whether each element of the domain is paired with exactly one element of the
range. For example, if given a graph, you could use the vertical line test; if a vertical line intersects the graph more
than once, then the relation that the graph represents is not a function.
56.Which point on the number line represents a number whose square is less than itself?
A A
B B
C C
D D
SOLUTION:
Consider the squares of all the numbers.
From the table, point B represents a number whose square is less than itself. Choice B is the correct answer.
Point Value Square
A–1.75 3.0625
B0.5 0.25
C1.33 1.7689
D2.85 8.1225
ANSWER:
B
57.Determine which of the following relations is a function.
F {(−3, 2), (4, 1), (−3, 5)}
G {(2, −1), (4, −1), (2, 6)}
H {(−3, −4), (−3, 6), (8, −2)}
J {(5, −1), (3, −2), (−2, −2)}
SOLUTION:
Considerthedomainandrangeforeachchoice.
From the table, choice J is the only one with three numbers in the domain row. The other rows have less, because
one of the domain members is repeated. Thus, {(5, −1), (3, −2), (−2, −2)} is the only relation where each element of
the domain is paired with exactly one element of the range. So, choice J is the correct answer.
Choice Domain Range
F {(−3, 2), (4, 1), (−3, 5)} –3, 4 2, 1, 5
G {(2, −1), (4, −1), (2, 6)} 2, 4 –1, 6
H {(−3, −4), (−3, 6), (8, −2)} –3, 8 –4, 6, –2
J {(5, −1), (3, −2), (−2, −2)} 5, 3, –2–1, –2
ANSWER:
J
58.GEOMETRY What is the value of x?
A 3 in
B 4 in.
C 5 in.
D 6 in.
SOLUTION:
Properties of similar triangles can be used to find x. The triangle with a hypotenuse of 4 is similar to the triangle with
a leg of 9. Set up a proportion to find x. Let the hypotenuse of each triangle be the numerator and let the
corresponding leg of each triangle be the denominator of each ratio.
So, x is 3 inches. Choices C and D are not possible since the leg of the triangle can not be longer than the
hypotenuse. Choice B would make the x the same length as the hypotenuse. That would only work if the hypotenuse
of the larger triangle equals the height of 9, which is not the case. Thus choice A is the correct answer.
ANSWER:
A
59.SHORT RESPONSE Camille made 16 out of 19 of her serves during her first volleyball game. She made 13 out of
16 of her serves during her second game. During which game did she make a greater percent of her serves?
SOLUTION:
First, each number of serves needs to be made into a percentage. Then, compare the percentages to see which is
greater.
In her first game, she made 84% of her serves, which is more than her second game of 81%.
ANSWER:
her first game
Solve each equation.
60.
SOLUTION:
ANSWER:
3
61.
SOLUTION:
ANSWER:
62.z = 32 + 4(−3)
SOLUTION:
ANSWER:
20
63.SCHOOL SUPPLIES The table shows the prices of some items Tom needs. If he needs 4 glue sticks, 10 pencils,
and 4 notebooks, write and evaluate an expression to determine Tom’scost.
SOLUTION:
To find the amount Tom will spend, multiply the price by the number of each item purchased. The sum of these will
give the total cost. Let g be the number of glue sticks, p the number of pencils, and n the number of notebooks.
Substitute 4 for g, 10 for p, and 4 for n.
ThetotalcostoftheschoolsuppliesTomneedsis$17.86.
ANSWER:
4($1.99) + 10(0.25) + (1.85) = 17.86, so the cost is $17.86.
Write a verbal expression for each algebraic expression.
64.4y + 2
SOLUTION:
The expression shows the sum of two terms, 4y and 2. Because the 4 and y are written next to each other, they are
being multiplied. So, the verbal expression four times y plus two can be used to describe the algebraic expression 4y
+ 2.
ANSWER:
Sample answer: four times y plus two
65.
SOLUTION:
Because andx are next to each other, they are being multiplied. So, the verbal expression two-thirds times x can
be used to describe the algebraic expression .
ANSWER:
Sample answer: two thirds times x
66.a2b + 5
SOLUTION:
The expression shows the sum of two terms, a2b and 5. Because a2 and b are next to each other, they are being
multiplied. The factor a2 represents a number raised to the second power. So, the verbal expression a squared times
b plus 5 can be used to describe the algebraic expression a2b + 5.
ANSWER:
Sample answer: a squared times b plus 5
Find the volume of each rectangular prism.
67.
SOLUTION:
Replace with 3.2, w with 2.2, and h with 5.4.
So, the volume of the rectangular prism is 38.016 cubic centimeters.
ANSWER:
38.016 cm3
68.
SOLUTION:
The length, width, and height of the solid are each in.
So, the volume of the rectangular prism is cubicinches.
ANSWER:
in.3
69.
SOLUTION:
Let = 180, w = 40, and h = 40.
So, the volume of the rectangular prism is 288,000 cubic millimeters.
ANSWER:
288,000 mm3
Evaluate each expression.
70.If x = 3, then 6x − 5 = ?.
SOLUTION:
ANSWER:
13
71.If n = −1, then 2n + 1 = ?.
SOLUTION:
ANSWER:
−1
72.If p = 4, then 3p + 4 = ?.
SOLUTION:
ANSWER:
16
73.If q = 7, then 7q − 9 = ?.
SOLUTION:
ANSWER:
40
74.If y = 10, then 8y − 15 = ? .
SOLUTION:
ANSWER:
65
75.If k = −11, then 4k + 6 = ? .
SOLUTION:
ANSWER:
−38
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1-7 Functions
