Unit 1A

Precalculus

Lesson 1: Domain, Range, and Interval Notation

A bounded set has an upper and lower limit, while in a set that is unbounded, there may be no upper limit, no lower limit, or neither an upper or a lower limit. Unbounded sets can be expressed using inequalities, set builder notation, or interval notation with the infinity sign, «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mo»§#8734;«/mo»«/mstyle»«/math».

Example 2

Represent the interval «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mfenced»«mrow»«mo»-«/mo»«mn»1«/mn»«mo»,«/mo»«mo»§#160;«/mo»«mo»§#8734;«/mo»«/mrow»«/mfenced»«/mstyle»«/math» on a number line. Express the interval as an inequality.

Solution

Recall less than and greater than symbols «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mfenced»«mrow»«mo»§#60;«/mo»«mo»,«/mo»«mo»§#160;«/mo»«mo»§#62;«/mo»«/mrow»«/mfenced»«/mstyle»«/math» correspond to open circles on number lines.

Inequality: «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»x«/mi»«mo»§#62;«/mo»«mo»-«/mo»«mn»1«/mn»«/mstyle»«/math»

Example 3

Graph the inequality «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mo»-«/mo»«mn»2«/mn»«mo»§#8805;«/mo»«mi»x«/mi»«/mrow»«/mstyle»«/math» and express it in interval notation.

Solution

Note an unbounded interval, going to «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mo»-«/mo»«mo»§#8734;«/mo»«/mrow»«/mstyle»«/math» or «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mo»§#8734;«/mo»«/mstyle»«/math» will always have curved parentheses, rather than square brackets, at the infinite end(s).

Interval Notation: «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mo»(«/mo»«mo»-«/mo»«mo»§#8734;«/mo»«mo»,«/mo»«mo»§#160;«/mo»«mo»-«/mo»«mn»2«/mn»«mo»]«/mo»«/mstyle»«/math»

Example 4

The graph of the function «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»y«/mi»«mo»=«/mo»«mi»f«/mi»«mfenced»«mi»x«/mi»«/mfenced»«/mrow»«/mstyle»«/math» is shown below. State the domain of «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»y«/mi»«mo»=«/mo»«mi»f«/mi»«mfenced»«mi»x«/mi»«/mfenced»«/mrow»«/mstyle»«/math» in set builder notation and interval notation.

Solution

There is an open circle at «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»x«/mi»«mo»=«/mo»«mo»-«/mo»«mn»2«/mn»«/mrow»«/mstyle»«/math». This means that point is not included in the domain of the function. There is a closed circle at «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»x«/mi»«mo»=«/mo»«mn»1«/mn»«/mrow»«/mstyle»«/math». This means that point is included in the function’s domain. The function is defined for «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»x«/mi»«mo»§#62;«/mo»«mo»-«/mo»«mn»2«/mn»«/mrow»«/mstyle»«/math» and «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»x«/mi»«mo»§#8804;«/mo»«mn»1«/mn»«/mrow»«/mstyle»«/math».

Set builder notation: «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mfenced open=¨{¨ close=¨}¨»«mrow»«mi»x«/mi»«mi mathvariant=¨normal¨»|«/mi»«mo»§#8722;«/mo»«mn»2«/mn»«mo»§#60;«/mo»«mi»x«/mi»«mo»§#8804;«/mo»«mn»1«/mn»«mi mathvariant=¨normal¨»,«/mi»«mo»§#160;«/mo»«mi»x«/mi»«mo»§#8712;«/mo»«mi mathvariant=¨normal¨»R«/mi»«/mrow»«/mfenced»«/mstyle»«/math»

Interval notation: «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mfenced close=¨]¨»«mrow»«mo»§#8722;«/mo»«mn»2«/mn»«mi mathvariant=¨normal¨»,«/mi»«mo»§#160;«/mo»«mn»1«/mn»«/mrow»«/mfenced»«/mstyle»«/math»