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Using Interval Notation

Module by: Pradnya Bhawalkar, Kim Johnston

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

Note: Your browser may not currently support MathML. See our browser support page for additional details. You can always view the correct math in the PDF version.

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

-52 5 2

Exercise 4

Figure 4
Figure 4 (1_4.png)

Solution

-∞-2⋃02⋃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: -23 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: -40 4 0

Range: 04 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: -44 4 4

Range: -4⋃-2⋃0⋃2 4 2 0 2

Exercise 20

Figure 20
Figure 20 (1_20.png)

Solution

Domain: -44 4 4

Range: 04 0 4

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