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FG2: Interval notation

Decorative Domains on an axis and how to write these domains mathematically

Often the domain of a function will be restricted to a subset of R. This subset is called an interval, and the end points are a and b.

Intervals

Often the domain of a function will be restricted to a subset of the set of real numbers, \(\mathbb{R}.\)

This subset is called an interval and the end points are \(a\) and \(b\) .

An interval may be represented on a real number line as follows:

Domain range from a to b inclusive

In inequality notation the above number line would be written as \(a\leq x\leq b.\)

In interval notation the above interval would be written as \([a,b]\).

Closed interval

Because the endpoints are included in the interval, this is called a closed interval and square brackets are used, eg. \(\left[2,5\right]\).

The end points on the real number line are represented by solid circles (or square brackets).

Open interval

If the endpoints are not included in the interval, this is called an open interval and curved brackets are used, eg. \(\left(2,5\right)\).

The end points on the real number line are represented by open circles (or curved brackets).

Domain range from a to b not including a and b

This is written in inequality notation as \(a<x<b\) . In interval notation as \(\left(a,b\right)\).

Domains on an axis and how to write these domains mathematically

In interval notation the smaller number is always written to the left; i.e.\([-3,5)\) not \((5,-3]\)

Note: the symbol \(\infty\) (infinity) is not a numeral.

\(\infty\) is the concept of continuing indefinitely to the right; \(-\infty\) is the concept of continuing indefinitely to the left.

Hence we cannot write \(\left[b,\infty\right]\) , \(\left[-\infty,a\right]\) or \(b\leq x\leq\infty\) etc.

Examples

  1. Write the following in inequality notation and graph on the real number line:
  1. \([-2,3)\)
    Inequality notation: \(-2\leq x<3\)
Domain range from -2 inclusive to 3 not inclusive
  1. \((-\infty,3]\)
    Inequality notation: \(x\leq3\)
Domain range from 3 inclusive to negative infinity
  1. Write the interval notation and inequality notation for the following line graphs:
Domain range from -5 non inclusive to 6 inclusive

Interval notation: \((-5,6]\)
Inequality notation: \(-5<x\leq6\)

Domain range from 10 inclusive to positive infinity

Interval notation: \([10,\infty)\)
Inequality notation: \(x\geq10\)

See Exercise 1.

Two intervals

Two (or more) subsets of \(R\), with end points \(a\) and \(b\), and \(c\) and \(d\), respectively, can also be represented on a real number line.

Examples

  1. Consider the line graph below:
Domain range from a to b inclusive and from c to d inclusive

This is written in interval notation as \([a,b]\cup[c,d]\). The symbol \(\cup\) means “in union with”. In inequality notation this may be written: \(a\leq x\leq b\) with \(c\leq x\leq d\) , or written as \(\left\{ x:a\leq x\leq b\right\} \cup\left\{ x:c\leq x\leq d\right\}\)

  1. Consider the line graph below:
Domain range from negative infinity 2 inclusive and from 5 not inclusive to 12 inclusive

This is written in interval notation as \((-\infty,2]\cup(5,12]\). In inequality notation this may be written: \(x\leq2\) with \(5<x\leq12\) , or written as \(\left\{ x:x\leq2\right\} \cup\left\{ x:5<x\leq12\right\}\).

See Exercises 2 and 3.

Exercises

  1. Write the following inequalities in interval notation and graph on a real number line:
    (a) \(1\leq x<10\)
    (b) \(-6\leq x<-4\)
    (c) \(x>5\)
  2. Write the following in interval notation and inequality notation:
Domain range from negative infinity to -5 inclusive
  1. Write the following in interval notation and inequality notation:
Domain range from -3 non inclusive to 0 non inclusive
  1. Write the following in interval notation and inequality notation:
Domain range from -1 inclusive to 4 non inclusive
  1. Graph the following on the real number line and write in inequality notation:
    (a) \(\left(-\infty,3\right)\cup(8,13]\)
    (b) \(\left[-1,4\right]\cup\left[6,9\right]\)
    (c) \((-\infty,3]\cup\left(6,\infty\right)\)

    1. \([1,10)\)
Domain range from 1 inclusive to 10 non inclusive

(b) \([-6,-4)\)

Domain range from -6 inclusive to -4 non inclusive

(c) \((5,\infty)\)

  1. \((-\infty,5]\) ; \(x\leq5\)
  2. \(\left(-3,0\right)\) ; \(-3<x<0\)
  3. \([-1,4)\) ; \(-1\leq x<4\)
    1. \(x<3\) with \(8<x\leq13\) or \(\left\{ x:x<3\right\} \cup\left\{ x:8<x\leq13\right\}\)
Domain range from -1inclusive to 4 inclusive and from 6 inclusive to 9 inclusive
  1. \(-1\leq x\leq4\) with \(6\leq x\leq9\) or \(\left\{ x:-1\leq x\leq4\right\} \cup\left\{ x:6\leq x\leq9\right\}\)
Domain range from -1 to 4 inclusive and 6 to 9 inclusive
  1. \(x\leq3\) with \(x>6\) this could also be written as \(\left\{ x:x\leq3\right\} \cup\left\{ x:x>6\right\}\)
Domain range from negative infinity to 3 inclusive and 6 non inclusive to positive infinity

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