AP Calculus AB and BC glossary

Interval notation

Interval notation describes a set of real numbers by its two endpoints: a square bracket includes the endpoint and a parenthesis excludes it. The interval from 2 to 5 with brackets contains both 2 and 5, the same interval with parentheses contains neither, and infinity always takes a parenthesis.

There are three shapes. Closed, $[a, b]$ , includes both endpoints. Open, $(a, b)$ , excludes both. Half-open, $[a, b)$ or $(a, b]$ , includes one. When a set has a gap in it you join the pieces with $\cup$ , which is how domains of rational functions get written: the domain of $\frac{1}{x - 3}$ is $(-\infty, 3) \cup (3, \infty)$ .

AP answers use this notation constantly: intervals where $f$ is increasing, intervals of concavity, and the interval of convergence of a power series. Convergence is the place where the bracket is graded hardest, because the ratio test only hands you the open interval. Each endpoint has to be substituted back and tested on its own before you can decide whether it earns a bracket or a parenthesis.

The mistake

Closing a bracket on a value the set does not contain. Writing $(0, \infty]$ is always wrong, since infinity is not a number and cannot be an element of anything. The same slip shows up on domains: the domain of $\ln x$ is $(0, \infty)$ , not $[0, \infty)$ , because $\ln 0$ is undefined. Check every endpoint by asking whether that exact value belongs to the set.

Appears in: Unit 1: Limits and Continuity