 ## FG2 Interval notation 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: 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). This is written in inequality notation as $$a<x<b$$ . In interval notation as $$\left(a,b\right)$$.

#### Examples 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: $$-2\leq x<3$$ 1. $$(-\infty,3]$$
Inequality notation: $$x\leq3$$ 1. Write the interval notation and inequality notation for the following line graphs: Interval notation: $$(-5,6]$$
Inequality notation: $$-5<x\leq6$$ 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: 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: 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: 1. Write the following in interval notation and inequality notation: 1. Write the following in interval notation and inequality notation: 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. $$[1,10)$$ (b) $$[-6,-4)$$ (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 $$\left\{ x:x<3\right\} \cup\left\{ x:8<x\leq13\right\}$$ 1. $$-1\leq x\leq4$$ with $$6\leq x\leq9$$ or $$\left\{ x:-1\leq x\leq4\right\} \cup\left\{ x:6\leq x\leq9\right\}$$ 1. $$x\leq3$$ with $$x>6$$ this could also be written as $$\left\{ x:x\leq3\right\} \cup\left\{ x:x>6\right\}$$ What's next... FG3 Inverse notation

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