Use notations to specify domain and range

In the previous examples, we used inequalities and lists to describe the domain of functions. We can also use inequalities, or other statements that might define sets of values or data, to describe the behavior of the variable in set-builder notation. For example, [latex]\left\{x|10\le x<30\right\}[/latex] describes the behavior of [latex]x[/latex] in set-builder notation. The braces { } are read as "the set of," and the vertical bar | is read as "such that," so we would read [latex]\left\{x|10\le x<30\right\}[/latex] as "the set of x-values such that 10 is less than or equal to [latex]x[/latex], and [latex]x[/latex] is less than 30."

The table below compares inequality notation, set-builder notation, and interval notation.

	Inequality Notation	Set-builder Notation	Interval Notation
	5 < h ≤ 10	{ h | 5 < h ≤ 10}	(5, 10]
	5 ≤ h < 10	{ h | 5 ≤ h < 10}	[5, 10]
	5 < h < 10	{ h | 5 < 10 }	(5, 10)
	h < 10	{ h | h < 10 }	( −∞, 10)
	h ≥ 10	{ h | h ≥ 10 }	[10, ∞ )
	All real numbers	ℝ	( −∞, ∞ )

To combine two intervals using inequality notation or set-builder notation, we use the word "or." As we saw in earlier examples, we use the union symbol, [latex]\cup [/latex], to combine two unconnected intervals. For example, the union of the sets [latex]\left\{2,3,5\right\}[/latex] and [latex]\left\{4,6\right\}[/latex] is the set [latex]\left\{2,3,4,5,6\right\}[/latex]. It is the set of all elements that belong to one or the other (or both) of the original two sets. For sets with a finite number of elements like these, the elements do not have to be listed in ascending order of numerical value. If the original two sets have some elements in common, those elements should be listed only once in the union set. For sets of real numbers on intervals, another example of a union is

Example 5: Describing Sets on the Real-Number Line

Describe the intervals of values shown in Figure 4 using inequality notation, set-builder notation, and interval notation.

Figure 4

Solution

To describe the values, [latex]x[/latex], included in the intervals shown, we would say, " [latex]x[/latex] is a real number greater than or equal to 1 and less than or equal to 3, or a real number greater than 5."

Inequality	[latex]1\le x\le 3\text{or}x>5[/latex]
Set-builder notation	[latex]\left\{x|1\le x\le 3\text{or}x>5\right\}[/latex]
Interval notation	[latex]\left[1,3\right]\cup \left(5,\infty \right)[/latex]

Remember that, when writing or reading interval notation, using a square bracket means the boundary is included in the set. Using a parenthesis means the boundary is not included in the set.

Try It 5

Given Figure 5, specify the graphed set in

1. words
2. set-builder notation
3. interval notation

Figure 5

Solution

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