Calculus How To


Interval Notation

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Open and Closed intervals >

Interval notation is a shorthand way to show:

  • The domain of a function (i.e., where the function is defined [1],
  • A range of values that make an inequality true,
  • Subsets of the real number line.

Interval Notation Reference List

The following image shows representations for inequalities, their respective interval notation, and how they appear on a graph.
The ∞ symbol is used to represent infinity; infinity is not a number, so it should never be paired with a square bracket when using interval notation.

  • Positive infinity (∞) indicates there isn’t an endpoint to the right of the number (on the number line).
  • Negative infinity (∞) indicates there isn’t an endpoint to the left of the number (on the number line).

Note that on a graph, a closed circle ● can be used instead of brackets [ ] and an open circle ○ can be used instead of parentheses ( ). 1
interval notation

Some General Tips

The smaller value of the interval goes first and the larger value of the interval goes second; the two values are separated by a comma. Parentheses indicate an open interval and square brackets a closed interval, although you can also have a mixture of the two, giving a half closed or half open interval. For example:

  • (2, 9) = an open interval from 2 to 9 (does not include 2 and 9 as endpoints).
  • [2, 9] = a closed interval from 2 to 9 (includes 2 and 9 as endpoints).
  • [2, 9) = a half-closed interval from 2 to 9 (includes 2 as an endpoint, but not 9).

Not sure what closed, open, and half-closed intervals are? Watch the video for an overview:

Closed Intervals, Open Intervals, Half Open, Half Closed
Watch this video on YouTube.

Can’t see the video? Click here.


If you have either infinity or negative infinity on either end of the interval, you always use a curve for that end. This will indicate that there is no definite endpoint in that direction, it keeps going and going.

References

[1] Domain and Range.
[2] Interval Notation. Retrieved November 4, 2021 from: https://math.osu.edu/sites/math.osu.edu/files/1148-interval-notations.pdf

CITE THIS AS:
Stephanie Glen. "Interval Notation" From CalculusHowTo.com: Calculus for the rest of us! https://www.calculushowto.com/interval-notation/
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