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2-1 Relations and Functions
State the domain and range of each relation. Then determine whether each relation is a function . If it is a
function, determine if it is one-to-one, onto, both, or neither.
11. SOLUTION: The members of the domain are the x-values of the relation while the members of the range are the y-values.
D = {–0.3, 0.4, 1.2}, R = {–6, –3, –1, 4}
1.2 is mapped to both –1 and 4. So the relation is not a function.
13. {(–3, –4), (–1, 0), (3, 0), (5, 3)}
SOLUTION: The members of the domain are the x-values of the relation while the members of the range are the y-values.
D = {–3, –1, 3, 5}; R = {–4, 0, 3}
Each element of the domain is paired with exactly one element in the range. So, the relation is a function.
The function is onto because each element of the range corresponds to an element of the domain.
CCSS STRUCTURE Graph each equation, and determine the domain and range. Determine whether the
equation is a function, is one-to-one, onto, both, or neither. Then state whether it is discrete or continuous.
15. SOLUTION: To graph, substitute values for x into the equation and solve for y. Draw a smooth curve through these points.
x
y = -3x + 2
0
1
2
3
-1
-2
-3
2
-1
-4
-7
5
8
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D = {–3, –1, 3, 5}; R = {–4, 0, 3}
Each element of the domain is paired with exactly one element in the range. So, the relation is a function.
2-1 Relations
and Functions
The function is onto because each element of the range corresponds to an element of the domain.
CCSS STRUCTURE Graph each equation, and determine the domain and range. Determine whether the
equation is a function, is one-to-one, onto, both, or neither. Then state whether it is discrete or continuous.
15. SOLUTION: To graph, substitute values for x into the equation and solve for y. Draw a smooth curve through these points.
x
y = -3x + 2
0
1
2
3
-1
-2
-3
2
-1
-4
-7
5
8
11
The members of the domain are the x-values of the relation while the members of the range are the y-values.
D = {all real numbers};
R = {all real numbers};
No vertical line intersects the graph in more than one point. So the equation is a function.
The function is both one-to-one and onto because each element of the domain is paired with a unique element of
the range and each element of the range corresponds to an element of the domain.
The domain has an infinite number of elements and the relation can be graphed with a solid straight line. So the
relation is continuous.
17. SOLUTION: To graph, substitute values for x into the equation and solve for y. Draw a smooth curve through these points.
x
y = 2x
2
0
0
1
2
2
8
3
18
-1
2
-2
8
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the range and each element of the range corresponds to an element of the domain.
The domain has an infinite number of elements and the relation can be graphed with a solid straight line. So the
2-1 Relations
and Functions
relation is continuous.
17. SOLUTION: To graph, substitute values for x into the equation and solve for y. Draw a smooth curve through these points.
x
y = 2x
2
0
1
2
3
-1
-2
-3
0
2
8
18
2
8
18
The members of the domain are the x-values of the relation while the members of the range are the y-values.
D = {all real numbers};
No vertical line intersects the graph in more than one point. So the equation is a function.
The function is not one-to-one because each element of the domain is not paired with a unique element of the range.
The function is not onto because the negative numbers are left unmapped.
The domain has an infinite number of elements and the relation can be graphed with a smooth curve. So the relation
is continuous.
19. SOLUTION: To graph, substitute values for x into the equation and solve for y. Draw a smooth curve through these points.
y = 4x 2 - 8
x
0
1
2
3
-1
-2
-3
-8
-4
8
28
-4
8
28
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The function is not onto because the negative numbers are left unmapped.
2-1 Relations
and
The domain
hasFunctions
an infinite number of elements and the relation can be graphed with a smooth curve. So the relation
is continuous.
19. SOLUTION: To graph, substitute values for x into the equation and solve for y. Draw a smooth curve through these points.
y = 4x 2 - 8
x
0
1
2
3
-1
-2
-3
-8
-4
8
28
-4
8
28
The members of the domain are the x-values of the relation while the members of the range are the y-values.
D = {all real numbers};
No vertical line intersects the graph in more than one point. So the equation is a function.
The function is not one-to-one because each element of the domain is not paired with a unique element of the range.
The function is not onto because the numbers less than –8 are left unmapped.
The domain has an infinite number of elements and the function can be graphed with a smooth curve. So the function
is continuous.
Evaluate each function.
21. SOLUTION: Replace x with –8.
23. DIVING The table below shows the pressure on a diver at various depths.
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Replace x with –8.
2-1 Relations and Functions
23. DIVING The table below shows the pressure on a diver at various depths.
a. Write a relation to represent the data.
b. Graph the relation.
c. Identify the domain and range. Is the relation discrete or continuous?
d. Is the relation a function? Explain your reasoning.
SOLUTION: a. Let the depth measurements be the domain and the pressure be the range.{(0, 1), (20, 1.6), (40, 2.2), (60, 2.8),
(80, 3.4), (100, 4)}
b. Plot each ordered pair from part a on the graph. Draw a straight line through the points. c. The depth begins at 0 and can increase indefinitely. The pressure begins at 1 and can increase indefinitely so the
domain and range are: D =
; R=
. The relation is continuous because the graph represents the
pressure at depths other than the given measures, such as 10 feet.
d. Each domain value is paired with only one range value. So the relation is a function.
Find each value if
and 25. SOLUTION: f (x) = 3x + 2
Replace x with 9.
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SOLUTION: Page 5
2-1 Relations and Functions
27. SOLUTION: Replace x with –6.
29. SOLUTION: Replace x with 8
.
31. SOLUTION: Replace x with
.
33. PODCASTS Chaz has a collection of 15 podcasts downloaded on his digital audio player. He decides to download 3
more podcasts each month. The function P(t) = 15 + 3t counts the number of podcasts P(t) he has after t months.
How many podcasts will he have after 8 months?
SOLUTION: Replace t with 8.
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2-1 Relations and Functions
33. PODCASTS Chaz has a collection of 15 podcasts downloaded on his digital audio player. He decides to download 3
more podcasts each month. The function P(t) = 15 + 3t counts the number of podcasts P(t) he has after t months.
How many podcasts will he have after 8 months?
SOLUTION: Replace t with 8.
After 8 months Chaz will have 39 podcasts.
35. CCSS CRITIQUE Omar and Madison are finding f (3d) for the function
correct? Explain your reasoning.
Is either of them SOLUTION: Sample answer: Both set up the equation correctly, substituting 3d for x in f (x). However, Madison did not square
the 3 before multiplying by –4 so Omar is correct.
37. REASONING If the graph of a relation crosses the y-axis at more than one point, is the relation sometimes,
always, or never a function? Explain your reasoning.
SOLUTION: Never; if the graph crosses the y-axis twice, then there will be two separate y-values that correspond to x = 0, which
violates the vertical line test.
39. REASONING Determine whether the following statement is true or false . Explain your reasoning.
If a function is onto, then it must be one-to-one as well.
SOLUTION: Sample answer: if a function is onto then each element of the range corresponds to an element of the domain. A
function that is one-to-one has each element of the domain paired to exactly one unique element of the range. The
statement is false; a function is onto and not one-to-one if all of the elements of the domain correspond to an element
of the range, but more than one element of the domain corresponds to the same element of the range.
41. Patricia’s swimming pool contains 19,500 gallons of water. She drains the pool at a rate of 6 gallons per minute.
Which of these equations represents the number of gallons of water g, remaining in the pool after m minutes?
A g = 19,500 – 6m
B g = 19,500 + 6m
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SOLUTION: Sample answer: if a function is onto then each element of the range corresponds to an element of the domain. A
function that is one-to-one has each element of the domain paired to exactly one unique element of the range. The
statement is
false;
a function is onto and not one-to-one if all of the elements of the domain correspond to an element
2-1 Relations
and
Functions
of the range, but more than one element of the domain corresponds to the same element of the range.
41. Patricia’s swimming pool contains 19,500 gallons of water. She drains the pool at a rate of 6 gallons per minute.
Which of these equations represents the number of gallons of water g, remaining in the pool after m minutes?
A g = 19,500 – 6m
B g = 19,500 + 6m
C D SOLUTION: Number of gallons of water in the pool = 19,500.
Patricia drains the water at a rate of 6 gallons per minute.
In m minutes, she can drain 6m gallons of water.
So the number of gallons of water remaining in the pool after m minutes is given by:
g = 19,500 – 6m
The correct choice is A.
43. GEOMETRY Which set of dimensions represents a triangle similar to the triangle shown below?
F 1 unit, 2 units, 3 units
G 7 units, 11 units, 12 units
H 10 units, 23 units, 24 units
J 20 units, 48 units, 52 units
SOLUTION: No common ratio can be found between the dimensions of the given triangle and the sets of dimensions given in the
choices F, G, H.
The dimensions of the triangle and the dimensions given in the choice J are in the ration 1:4.
So the correct choice is J.
Solve each inequality.
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choices F, G, H.
The dimensions of the triangle and the dimensions given in the choice J are in the ration 1:4.
2-1 Relations
and Functions
So the correct choice is J.
Solve each inequality.
45. SOLUTION: 47. SOLUTION: This implies:
49. SALES Ling can spend no more than $120 at the summer sale of a department store. She wants to buy shirts on
sale for $15 each. Write and solve an inequality to determine the number of shirts she can buy.
SOLUTION: Let x be the number of shirts Ling can buy.
Each shirt costs $15.
So:
Ling can buy a maximum of 8 shirts.
Solve each equation. Check your solutions.
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2-1 Relations
and Functions
Ling can buy a maximum of 8 shirts.
Solve each equation. Check your solutions.
51. SOLUTION: This implies:
Simplify each expression.
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