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Using Interval Notation

Module by: Pradnya Bhawalkar, Kim Johnston. E-mail the authors

Summary: An introductory explanation on how to use interval notation (versus set notation)

Interval notation is another method for writing domain and range.

In set builder notation braces (curly parentheses {} ) and variables are used to express the domain and range. Interval notation is often considered more efficient.

In interval notation, there are only 5 symbols to know:

  • Open parentheses ( )
  • Closed parentheses [ ]
  • Infinity ∞
  • Negative Infinity −∞
  • Union Sign ∪

To use interval notation:

Use the open parentheses ( ) if the value is not included in the graph. (i.e. the graph is undefined at that point... there's a hole or asymptote, or a jump)

If the graph goes on forever to the left, the domain will start with ( −∞. If the graph travels downward forever, the range will start with ( −∞. Similarly, if the graph goes on forever at the right or up, end with ∞)

Use the brackets [ ] if the value is part of the graph.

Whenever there is a break in the graph, write the interval up to the point. Then write another interval for the section of the graph after that part. Put a union sign between each interval to "join" them together.

Now for some practice so you can see if any of this makes sense.

Write the following using interval notation:

Exercise 1

Figure 1
Figure 1 (1_1.png)

Solution

0 ∞ 0

Exercise 2

Figure 2
Figure 2 (1_2.png)

Solution

−∞ −2 ∪ 1 ∞ 2 1

Exercise 3

Figure 3
Figure 3 (1_3.png)

Solution

−5 2 5 2

Exercise 4

Figure 4
Figure 4 (1_4.png)

Solution

−∞ −2 ∪ 0 2 ∪ 4 ∞ 2 0 2 4

Exercise 5

Figure 5
Figure 5 (1_5.png)

Solution

−1∪ 3 ∞ 1 3

Exercise 6

Figure 6
Figure 6 (1_6.png)

Solution

−∞ 3 3

Write the domain and range of the following in interval notation:

Exercise 7

Figure 7
Figure 7 (1_7.png)

Solution

Domain: −2∪0∪2∪3 2 0 2 3

Range: −1∪1∪2∪3 1 1 2 3

Exercise 8

Figure 8
Figure 8 (1_8.png)

Solution

Domain: −∞ ∞

Range: ∞ 1 ∪4 1 4

Exercise 9

Figure 9
Figure 9 (1_9.png)

Solution

Domain: −∞ ∞

Range: −∞ ∞

Exercise 10

Figure 10
Figure 10 (1_10.png)

Solution

Domain: −∞ ∞

Range: 1∪3 1 3

Exercise 11

Figure 11
Figure 11 (1_11.png)

Solution

Domain: −∞ ∞

Range: −2 3 2 3

Exercise 12

Figure 12
Figure 12 (1_12.png)

Solution

Domain: −∞ ∞

Range: 1∪3 1 3

Exercise 13

Figure 13
Figure 13 (1_13.png)

Solution

Domain: −4 0 4 0

Range: 0 4 0 4

Exercise 14

Figure 14
Figure 14 (1_14.png)

Solution

Domain: −∞ ∞

Range: −∞ 4 4

Exercise 15

Figure 15
Figure 15 (1_15.png)

Solution

Domain: −∞ ∞

Range: 0 ∞ 0

Exercise 16

Figure 16
Figure 16 (1_16.png)

Solution

Domain: 0 ∞ 0

Range: −∞ ∞

Exercise 17

Figure 17
Figure 17 (1_17.png)

Solution

Domain: −3∪−2∪ −1 ∞ 3 2 1

Range: 0∪1∪2 0 1 2

Exercise 18

Figure 18
Figure 18 (1_18.png)

Solution

Domain: −∞ ∞

Range: 4 4

Exercise 19

Figure 19
Figure 19 (1_19.png)

Solution

Domain: −4 4 4 4

Range: −4∪−2∪0∪2 4 2 0 2

Exercise 20

Figure 20
Figure 20 (1_20.png)

Solution

Domain: −4 4 4 4

Range: 0 4 0 4

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