Download CCSS STRUCTURE State the domain and range of each relation

2-1 Relations and Functions
CCSS STRUCTURE 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.
2. SOLUTION: The members of the domain are the x-values of the relation while the members of the range are the y-values.
D = {–2, 1, 4}, R = {–1, 2, 3, 5};
The relation is not a function because 1 is mapped to both 2 and 5.
4. BASKETBALL The table shows the average points per game for Dwayne Wade of the Miami Heat for four
years.
a. Assume that the ages are the domain. Identify the domain and range.
b. Write a relation of ordered pairs for the data.
c. State whether the relation is discrete or continuous.
d. Graph the relation. Is this relation a function?
SOLUTION: a. Since the ages are the domain, the average points per game are the range.
D = {24, 25, 26, 27}, R = {24.6, 27.2, 27.4, 30.2}
b. In writing ordered pairs for the relation, the members of the domain are the x-values and the members of the
range are the y -values. {(24, 27.2), (25, 27.4), (26, 24.6), (27, 30.2)}
c. The domain is a set of individual points. So the relation is discrete.
d. The relation is a function as each element of the domain is paired with exactly one element of the range.
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D = {–2, 1, 4}, R = {–1, 2, 3, 5};
2-1 Relations
and
Functions
The relation
is not
a function because 1 is mapped to both 2 and 5.
4. BASKETBALL The table shows the average points per game for Dwayne Wade of the Miami Heat for four
years.
a. Assume that the ages are the domain. Identify the domain and range.
b. Write a relation of ordered pairs for the data.
c. State whether the relation is discrete or continuous.
d. Graph the relation. Is this relation a function?
SOLUTION: a. Since the ages are the domain, the average points per game are the range.
D = {24, 25, 26, 27}, R = {24.6, 27.2, 27.4, 30.2}
b. In writing ordered pairs for the relation, the members of the domain are the x-values and the members of the
range are the y -values. {(24, 27.2), (25, 27.4), (26, 24.6), (27, 30.2)}
c. The domain is a set of individual points. So the relation is discrete.
d. The relation is a function as each element of the domain is paired with exactly one element of the range.
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.
6. SOLUTION: To graph the equation, substitute different values of x in the equation and solve for y. Then connect the points.
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x
y = -4x - 2
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2-1 Relations and Functions
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.
6. SOLUTION: To graph the equation, substitute different values of x in the equation and solve for y. Then connect the points.
x
y = -4x - 2
0
1
2
3
-1
-2
-3
-2
-6
-10
-14
2
6
10
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 graph 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 correspond to an element of the domain.
The domain has an infinite number of elements and the relation can be graphed using a straight line. So the relation is
continuous.
8. SOLUTION: The graph of the equation is a vertical line through (7, 0). Manual - Powered by Cognero
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In this equation x is always 7 for any value of y.
D = {7}; R = {all real numbers};
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the range and each element of the range correspond to an element of the domain.
The domain has an infinite number of elements and the relation can be graphed using a straight line. So the relation is
2-1 Relations
continuous.and Functions
8. SOLUTION: The graph of the equation is a vertical line through (7, 0). In this equation x is always 7 for any value of y.
D = {7}; R = {all real numbers};
The only element in the domain is mapped to all the elements in the range. So it is not a function.
The domain has a finite number (1) of elements, so the relation is not continuous.
Evaluate each function.
10. SOLUTION: Replace x with 5.
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.
12. SOLUTION: The members of the domain are the x-values of the relation while the members of the range are the y-values.
D = {–8, 2, 4}; R = {–6, –4, 14};
–8 is mapped to both –4 and 14. So the relation is not a function.
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14. POLITICS The table below shows the population of several states and the number of U.S. representatives fromPage 4
those states.
a. Make a graph of the data with population on the horizontal axis and representatives on the vertical axis.
The members of the domain are the x-values of the relation while the members of the range are the y-values.
D = {–8, 2, 4}; R = {–6, –4, 14};
2-1 Relations
and Functions
–8 is mapped to both –4 and 14. So the relation is not a function.
14. POLITICS The table below shows the population of several states and the number of U.S. representatives from
those states.
a. Make a graph of the data with population on the horizontal axis and representatives on the vertical axis.
b. Identify the domain and range.
c. Is the relation discrete or continuous?
d. Does the graph represent a function? Explain your reasoning.
SOLUTION: a. Scale each axis of the graph by 5. Since population is on the horizontal axis, these are the x-values of the relation
or the domain. The number of representatives is the range. Plot the data from the table as ordered pairs on the
graph.
b. The members of the domain are the x-values of the relation while the members of the range are the y-values. D
= {8.07, 12.44, 16.03, 19.00, 20.90, 33.93}; R={13, 19, 25, 29, 32, 53}
c. The domain is a set of individual points. So the relation is discrete.
d. The relation is a function because each domain value is paired with only one range value.
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.
16. SOLUTION: To graph,
values
for x into the equation and solve for y. Draw a smooth curve through these points.
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y = 0.5x - 3
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= {8.07, 12.44, 16.03, 19.00, 20.90, 33.93}; R={13, 19, 25, 29, 32, 53}
c. The domain is a set of individual points. So the relation is discrete.
2-1 Relations
and Functions
d. The relation is a function because each domain value is paired with only one range value.
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.
16. SOLUTION: To graph, substitute values for x into the equation and solve for y. Draw a smooth curve through these points.
x
y = 0.5x - 3
0
1
2
3
-1
-2
-3
-3
-2.5
-2
-1.5
-3.5
-5
-4.5
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.
18. SOLUTION: To graph, substitute values for x into the equation and solve for y. Draw a smooth curve through these points.
x
y = -5x
2
0
0
1
-5
2
-20
3
-45
-1
-5
-2
-20
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-3
-45
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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.
18. SOLUTION: To graph, substitute values for x into the equation and solve for y. Draw a smooth curve through these points.
x
y = -5x
0
1
2
3
-1
-2
-3
2
0
-5
-20
-45
-5
-20
-45
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 positive 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.
20. SOLUTION: To graph, substitute values for x into the equation and solve for y. A few of the points on the graph are (0, -1), (1, 4), (-1, 2),
, (2, -25), and (-2, 23). Draw a smooth curve through these points.
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The members of the domain are the x-values of the relation while the members of the range are the y-values.
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The function is not onto because the positive 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.
20. SOLUTION: To graph, substitute values for x into the equation and solve for y. A few of the points on the graph are (0, -1), (1, 4), (-1, 2),
, (2, -25), and (-2, 23). Draw a smooth curve through these points.
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 smooth curve. So the relation
is continuous.
Evaluate each function.
22. SOLUTION: Replace x with 2.5.
Find each value if
and 24. SOLUTION: f (x) = 3x + 2
Replace x with –5.
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Replace x with 2.5.
2-1 Relations and Functions
Find each value if
and 24. SOLUTION: f (x) = 3x + 2
Replace x with –5.
26. SOLUTION: Replace x with –3.
.
28. SOLUTION: Replace x with 3.
30. SOLUTION: Replace x with
.
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2-1 Relations and Functions
30. SOLUTION: Replace x with
.
32. SOLUTION: Replace x with
.
34. MULTIPLE REPRESENTATIONS In this problem you will investigate one-to-one and onto functions.
a. GRAPHICAL Graph each function on a separate graphing calculator screen.
b. TABULAR Use the graphs to create a table showing the number of times a horizontal line could intersect the
graph of each function. List all possibilities.
c. ANALYTICAL For a function to be one-to-one, a horizontal line on the graph of the function can intersect the
function at most once. Which functions meet this condition? Which do not? Explain your reasoning.
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d. ANALYTICAL For a function to be onto, every possible horizontal line on the graph of the function must
intersect the function at least once. Which functions meet this condition? Which do not? Explain your reasoning.
b. TABULAR Use the graphs to create a table showing the number of times a horizontal line could intersect the
graph of each
List all possibilities.
2-1 Relations
andfunction.
Functions
c. ANALYTICAL For a function to be one-to-one, a horizontal line on the graph of the function can intersect the
function at most once. Which functions meet this condition? Which do not? Explain your reasoning.
d. ANALYTICAL For a function to be onto, every possible horizontal line on the graph of the function must
intersect the function at least once. Which functions meet this condition? Which do not? Explain your reasoning.
e . GRAPHICAL Create a table showing whether each function is one-to-one and/or onto.
SOLUTION: a. f (x) = x2
g(x) = 2x
h(x) = x 3 - 3x 2 - 5x + 6
j(x) = x 3
b.
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2-1 Relations
and Functions
b.
c. Placing a pencil on each graph so it’s parallel to the x-axis and then moving it straight up and down, only g(x) and
j(x) intersect the pencil line once at a time so they are one-to-one, and f (x) and h(x) are not
.
d. A horizontal line on the graphs of h(x) and j (x) will intersect the graph more than once so they are onto, and f (x)
and g(x) are not.
e. 36. CHALLENGE Consider the functions f (x) and
and g(a) = 33, while f (b) = 31 and g(b) = 51. If a
= 5 and b = 8, find two possible functions to represent f (x) and g(x).
SOLUTION: Sample answer: Organize the given information into a table. f (x)
g(x)
f (a) = 19
g(a) = 33
f (b) = 31
g(b) = 51
a = 5, b = 8
Analyze the information given about f (x).
f (a) = 19
f (5) = 19
f (x)
4(5) = 20
f (b) = 31
f (8) = 31
4(8) = 32
f (x) = 4x – 1
f (a) = 4(5) – 1 = 20
f (x) = 4x – 1
f (b) = 4(8) – 1 = 31
If the values of a and b are multiplied by 4, the product is one more than the value of f (a) and f (b). Next, analyze the information given about g(x).
g(x)
g(a) = 33
g(5) = 33
6(5) = 30
g(x) = 6x + 3
g(a) = 6(5) + 3 = 33
g(b) = 51
g(8) = 51
6(8) = 48
g(x) = 4x + 3
g(b) = 6(8) + 3 = 51
If the values of a and b are multiplied by 6, the product is three less than the value of g(a) and g(b).
So the functions are:
.
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38. OPEN ENDED Graph a relation that can be used to represent each of the following.
a. the height of a baseball that is hit into the outfield
b. the speed of a car that travels to the store, stopping at two lights along the way
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2-1 Relations and Functions
36. CHALLENGE Consider the functions f (x) and
and g(a) = 33, while f (b) = 31 and g(b) = 51. If a
= 5 and b = 8, find two possible functions to represent f (x) and g(x).
SOLUTION: Sample answer: Organize the given information into a table. f (x)
g(x)
f (a) = 19
g(a) = 33
f (b) = 31
g(b) = 51
a = 5, b = 8
Analyze the information given about f (x).
f (a) = 19
f (5) = 19
f (x)
4(5) = 20
f (b) = 31
f (8) = 31
4(8) = 32
f (x) = 4x – 1
f (a) = 4(5) – 1 = 20
f (x) = 4x – 1
f (b) = 4(8) – 1 = 31
If the values of a and b are multiplied by 4, the product is one more than the value of f (a) and f (b). Next, analyze the information given about g(x).
g(x)
g(a) = 33
g(5) = 33
6(5) = 30
g(x) = 6x + 3
g(a) = 6(5) + 3 = 33
g(b) = 51
g(8) = 51
6(8) = 48
g(x) = 4x + 3
g(b) = 6(8) + 3 = 51
If the values of a and b are multiplied by 6, the product is three less than the value of g(a) and g(b).
So the functions are:
.
38. OPEN ENDED Graph a relation that can be used to represent each of the following.
a. the height of a baseball that is hit into the outfield
b. the speed of a car that travels to the store, stopping at two lights along the way
c. the height of a person from age 5 to age 80
d. the temperature on a typical day from 6 A.M. to 11 P.M.
SOLUTION: a. Sample answer: let the x-axis be the time the ball is in the air and the y-axis be the height of the ball. The height of
the ball is zero when the time is zero. Once the ball is thrown, the height will reach a maximum point and then
decrease eventually landing on the ground.
b. Sample answer: let the x-axis be the time the car is being driven and the y-axis be the distance the car has
traveled. At each stop light, time increases but distance is constant. eSolutions Manual - Powered by Cognero
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g(b) = 51
g(8) = 51
6(8) = 48
g(x) = 4x + 3
g(b) = 6(8) + 3 = 51
If the values
ofFunctions
a and b are multiplied by 6, the product is three less than the value of g(a) and g(b).
2-1 Relations
and
So the functions are:
.
38. OPEN ENDED Graph a relation that can be used to represent each of the following.
a. the height of a baseball that is hit into the outfield
b. the speed of a car that travels to the store, stopping at two lights along the way
c. the height of a person from age 5 to age 80
d. the temperature on a typical day from 6 A.M. to 11 P.M.
SOLUTION: a. Sample answer: let the x-axis be the time the ball is in the air and the y-axis be the height of the ball. The height of
the ball is zero when the time is zero. Once the ball is thrown, the height will reach a maximum point and then
decrease eventually landing on the ground.
b. Sample answer: let the x-axis be the time the car is being driven and the y-axis be the distance the car has
traveled. At each stop light, time increases but distance is constant. c. Sample answer: let the x-axis be the age of the person and the y-axis be the height. Height increases as a child,
then more steeply, finally leveling off and remaining constant.
d. Sample answer: let the x-axis be the hours from 6 to 11 and the y-axis be the temperature. On a typical day, the
morning is cool while the temperature gradually warms up to reach a maximum. The temperature gradually
decreases as the sun sets.
40. WRITING IN MATH Explain why the vertical line test can determine if a relation is a function.
SOLUTION: Sample answer: A relation is a function if each x-value only pairs with one y-value. If the vertical line test fails then
there is an x-value that pairs with more than one y-value, so the relation is not a function.
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42. SHORT RESPONSE Look at the pattern below.
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2-1 Relations and Functions
40. WRITING IN MATH Explain why the vertical line test can determine if a relation is a function.
SOLUTION: Sample answer: A relation is a function if each x-value only pairs with one y-value. If the vertical line test fails then
there is an x-value that pairs with more than one y-value, so the relation is not a function.
42. SHORT RESPONSE Look at the pattern below.
If the pattern continues, what will the next term be?
SOLUTION: Each term of the pattern is obtained by adding
to the previous term.
Next term =
44. ACT/SAT If
A. 1
2
B. x + 1
which expression is equal to g(x + 1)?
2
C. x + 2x + 1
2
D. x – x
2
E. x + x + 1
SOLUTION: Replace x by x + 1.
The correct choice is C.
Solve each inequality.
46. SOLUTION: 48. CLUBS Mr. Willis is starting a chess club at his high school. He sent the advertisement at the right to all of the
homerooms. Write an absolute value inequality representing the situation.
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2-1 Relations and Functions
48. CLUBS Mr. Willis is starting a chess club at his high school. He sent the advertisement at the right to all of the
homerooms. Write an absolute value inequality representing the situation.
SOLUTION: Let x represent the number of members.
Solve each equation. Check your solutions.
50. SOLUTION: This implies:
52. SOLUTION: This implies:
Simplify each expression.
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2-1 Relations and Functions
Simplify each expression.
54. SOLUTION: Solve each equation. Check your solutions.
56. SOLUTION: Substitute x = 6 in the original equation.
So the solution is x = 6.
58. SOLUTION: Substitute x = –2 in the original equation.
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So the solution is x = –2.
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2-1 Relations
and is
Functions
So the solution
x = 6.
58. SOLUTION: Substitute x = –2 in the original equation.
So the solution is x = –2.
60. SOLUTION: Substitute x = 3 in the original equation.
So the solution is x = 3.
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Substitute c = 6 in the original equation.
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2-1 Relations
and Functions
So the solution is x = 3.
62. SOLUTION: Substitute c = 6 in the original equation.
So the solution is c = 6.
64. SOLUTION: Substitute y = 3 in the equation.
So the solution is y = 3.
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