Algebra 1 · Unit 2
Read the domain and the range from a discrete graph or a continuous graph.
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Discrete graphs
Domain is the set of x-values. Range is the set of y-values.
List the values from least to greatest. Do not repeat a value.
Write them in braces, such as {−2, 0, 3}.
Continuous graphs
Write a connected graph in interval notation.
A filled dot is included. An open circle is not. An arrow means that end is infinity.
If every point has the same y-value, the range is that one value in braces, such as {2}.
Gaps
An open circle on a line is a missing point. Leave that number out.
Two separate pieces are joined with ∪.
A graph can have a domain and a range even when it is not a function.
List the x-values and the y-values from least to greatest. Do not repeat a value.
Domain is the set of these values. List them from least to greatest. Do not repeat a value.
Values: 0, −3, 2, −1
From least to greatest, with no repeats: −3, −1, 0, 2
Domain = {−3, −1, 0, 2}
Range is the set of these values. List them from least to greatest. Do not repeat a value.
Values: 4, 1, 1, −2
From least to greatest, with no repeats: −2, 1, 4
Range = {−2, 1, 4}
Answer: Domain {−3, −1, 0, 2}; Range {−2, 1, 4}
Domain is the set of these values. List them from least to greatest. Do not repeat a value.
Values: 3, −2, 1, 4
From least to greatest, with no repeats: −2, 1, 3, 4
Domain = {−2, 1, 3, 4}
Range is the set of these values. List them from least to greatest. Do not repeat a value.
Values: −1, 0, 4, 2
From least to greatest, with no repeats: −1, 0, 2, 4
Range = {−1, 0, 2, 4}
Answer: Domain {−2, 1, 3, 4}; Range {−1, 0, 2, 4}
Domain is the set of these values. List them from least to greatest. Do not repeat a value.
Values: −2, 2, −4, 0
From least to greatest, with no repeats: −4, −2, 0, 2
Domain = {−4, −2, 0, 2}
Range is the set of these values. List them from least to greatest. Do not repeat a value.
Values: 3, −1, 3, 3
From least to greatest, with no repeats: −1, 3
Range = {−1, 3}
Answer: Domain {−4, −2, 0, 2}; Range {−1, 3}
Domain is the set of these values. List them from least to greatest. Do not repeat a value.
Values: 2, −3, 4, −1, 0
From least to greatest, with no repeats: −3, −1, 0, 2, 4
Domain = {−3, −1, 0, 2, 4}
Range is the set of these values. List them from least to greatest. Do not repeat a value.
Values: 4, −2, 1, 4, −2
From least to greatest, with no repeats: −2, 1, 4
Range = {−2, 1, 4}
Answer: Domain {−3, −1, 0, 2, 4}; Range {−2, 1, 4}
Domain is the set of these values. List them from least to greatest. Do not repeat a value.
Values: 1, 4, −2, 3
From least to greatest, with no repeats: −2, 1, 3, 4
Domain = {−2, 1, 3, 4}
Range is the set of these values. List them from least to greatest. Do not repeat a value.
Values: 2, 0, −4, 2
From least to greatest, with no repeats: −4, 0, 2
Range = {−4, 0, 2}
Answer: Domain {−2, 1, 3, 4}; Range {−4, 0, 2}
Domain is the set of these values. List them from least to greatest. Do not repeat a value.
Values: 0, −4, 3, 1, −2
From least to greatest, with no repeats: −4, −2, 0, 1, 3
Domain = {−4, −2, 0, 1, 3}
Range is the set of these values. List them from least to greatest. Do not repeat a value.
Values: −3, 1, 4, 1, −3
From least to greatest, with no repeats: −3, 1, 4
Range = {−3, 1, 4}
Answer: Domain {−4, −2, 0, 1, 3}; Range {−3, 1, 4}
Use interval notation. A single value goes in braces.
The line has arrows on both ends.
It is not vertical, so every x-value is included.
Domain = (−∞, ∞)
It is not horizontal, so every y-value is included.
Range = (−∞, ∞)
In set-builder notation, the domain is {x | −∞ < x < ∞}.
The range is {y | −∞ < y < ∞}.
Answer: Domain (−∞, ∞); Range (−∞, ∞)
The line has arrows on both ends, so x continues forever.
Domain = (−∞, ∞)
Every point has y = −2.
Range = {−2}
Answer: Domain (−∞, ∞); Range {−2}
The segment runs from (−4, −2) to (3, 1).
Left end x = −4 is included. Right end x = 3 is included.
Domain = [−4, 3]
Low y-value −2 is included. High y-value 1 is included.
Range = [−2, 1]
Answer: Domain [−4, 3]; Range [−2, 1]
The ray starts at x = 1, which is included. The arrow goes right, so x continues forever.
Domain = [1, ∞)
The ray starts at y = 3, which is included. The arrow goes down, so y continues forever.
Range = (−∞, 3]
Answer: Domain [1, ∞); Range (−∞, 3]
Arrows on both sides mean x continues forever.
Domain = (−∞, ∞)
The lowest point is y = −3, and that point is included. The arrows go up forever.
Range = [−3, ∞)
Answer: Domain (−∞, ∞); Range [−3, ∞)
The segment runs from x = −2 to x = 3.
The left end is included. The right end is included.
Domain = [−2, 3]
Every point has y = 4.
Range = {4}
Answer: Domain [−2, 3]; Range {4}
An open circle is not included.
The segment runs from (−4, −2) to (2, 3).
Left end x = −4 is not included. Right end x = 2 is included.
Domain = (−4, 2]
Low y-value −2 is not included. High y-value 3 is included.
Range = (−2, 3]
Answer: Domain (−4, 2]; Range (−2, 3]
The ray starts at x = 0, which is not included. The arrow goes right, so x continues forever.
Domain = (0, ∞)
The ray starts at y = 2, which is not included. The arrow goes up, so y continues forever.
Range = (2, ∞)
Answer: Domain (0, ∞); Range (2, ∞)
The segment runs from (−3, 4) to (1, −2).
Left end x = −3 is not included. Right end x = 1 is not included.
Domain = (−3, 1)
Low y-value −2 is not included. High y-value 4 is not included.
Range = (−2, 4)
Answer: Domain (−3, 1); Range (−2, 4)
The ray starts at x = −1, which is included. The arrow goes right, so x continues forever.
Domain = [−1, ∞)
The ray starts at y = −2, which is included. The arrow goes down, so y continues forever.
Range = (−∞, −2]
Answer: Domain [−1, ∞); Range (−∞, −2]
The ray starts at x = 2, which is not included. The arrow goes left, so x continues forever.
Domain = (−∞, 2)
The ray is horizontal, so every point has y = −3.
Range = {−3}
Answer: Domain (−∞, 2); Range {−3}
The segment runs from (−2, 1) to (3, 4).
Left end x = −2 is included. Right end x = 3 is not included.
Domain = [−2, 3)
Low y-value 1 is included. High y-value 4 is not included.
Range = [1, 4)
Answer: Domain [−2, 3); Range [1, 4)
Leave out a hole. Join separate pieces with ∪.
There is an open circle at (2, 2), so 2 is not included.
The line continues on both sides.
Domain = (−∞, 2) ∪ (2, ∞)
The missing y-value is 2 as well, because the open point is (2, 2) and the line is y = x.
Range = (−∞, 2) ∪ (2, ∞)
Answer: Domain (−∞, 2) ∪ (2, ∞); Range (−∞, 2) ∪ (2, ∞)
The circle reaches x = −3 on the left and x = 3 on the right. Both points are on the graph.
Domain = [−3, 3]
The circle reaches y = −3 and y = 3. Both points are on the graph.
Range = [−3, 3]
Answer: Domain [−3, 3]; Range [−3, 3]
There are two separate pieces.
First segment: x runs from −4 to −1, both included. y runs from −2 to 1, both included.
Second segment: x runs from 2 to 4, both included. y runs from 2 to 4, both included.
The pieces do not touch. Join them with ∪.
Domain = [−4, −1] ∪ [2, 4]
Range = [−2, 1] ∪ [2, 4]
Answer: Domain [−4, −1] ∪ [2, 4]; Range [−2, 1] ∪ [2, 4]
Domain is the set of these values. List them from least to greatest. Do not repeat a value.
Values: −3, −1, −1, 0, 3, 3, 4
From least to greatest, with no repeats: −3, −1, 0, 3, 4
Domain = {−3, −1, 0, 3, 4}
Range is the set of these values. List them from least to greatest. Do not repeat a value.
Values: 2, −2, 4, 2, 0, 4, −2
From least to greatest, with no repeats: −2, 0, 2, 4
Range = {−2, 0, 2, 4}
Answer: Domain {−3, −1, 0, 3, 4}; Range {−2, 0, 2, 4}
The graph is a vertical line at x = −2.
Domain = {−2}
Arrows go up and down, so y continues forever.
Range = (−∞, ∞)
Answer: Domain {−2}; Range (−∞, ∞)
Top segment: y = 1 from x = −4 to x = −1, both included.
Bottom segment: y = −2 from x = 1 to x = 4, both included.
Domain = [−4, −1] ∪ [1, 4]
The y-values are 1 and −2.
From least to greatest, with no repeats: −2, 1.
Range = {−2, 1}
Answer: Domain [−4, −1] ∪ [1, 4]; Range {−2, 1}
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