L1 Domain, Range, and Interval Notation - Part 4
Completion requirements
Unit 1A
Precalculus
Lesson 1: Domain, Range, and Interval Notation
Interval Notation
Consider a student whose daily job it is to deliver flyers to the houses along his street. He must deliver the flyers to all of the houses that lie «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mn»5«/mn»«mo»§#160;«/mo»«mi»km«/mi»«/mrow»«/mstyle»«/math» to the west and «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mn»5«/mn»«mo»§#160;«/mo»«mi»km«/mi»«/mrow»«/mstyle»«/math» to the east of his house. It is pretty straightforward to represent this situation using a number line.
Using a number line is just one way this route can be represented. Another way is to use an inequality. If «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»x«/mi»«/mstyle»«/math» represents the
distance from his house, the route can be written as «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mo»§#8722;«/mo»«mn»5«/mn»«mo»§#8804;«/mo»«mi»x«/mi»«mo»§#8804;«/mo»«mn»5«/mn»«/mrow»«/mstyle»«/math».
A different method of representing the same inequality is to use interval notation. Using interval notation, the route can be written as «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mfenced open=¨[¨ close=¨]¨»«mrow»«mo»-«/mo»«mn»5«/mn»«mo»,«/mo»«mo»§#160;«/mo»«mn»5«/mn»«/mrow»«/mfenced»«/mstyle»«/math».
In interval notation, square brackets represent included values and round brackets represent excluded values. The numbers within the brackets contain the lowest and highest allowable values of the interval.
When graphing on a number line, a solid dot indicates inclusion (the number at the solid dot is included in the set) and an open circle indicates exclusion (the number at the open circle is excluded from the set).
A different method of representing the same inequality is to use interval notation. Using interval notation, the route can be written as «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mfenced open=¨[¨ close=¨]¨»«mrow»«mo»-«/mo»«mn»5«/mn»«mo»,«/mo»«mo»§#160;«/mo»«mn»5«/mn»«/mrow»«/mfenced»«/mstyle»«/math».
In interval notation, square brackets represent included values and round brackets represent excluded values. The numbers within the brackets contain the lowest and highest allowable values of the interval.
When graphing on a number line, a solid dot indicates inclusion (the number at the solid dot is included in the set) and an open circle indicates exclusion (the number at the open circle is excluded from the set).
Use interval notation to state the interval represented on the number line.
The open circles indicate that «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mo»-«/mo»«mn»3«/mn»«/mrow»«/mstyle»«/math» and «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle
mathsize=¨14px¨»«mrow»«mo»+«/mo»«mn»3«/mn»«/mrow»«/mstyle»«/math» are not included in the interval.
The interval represented above can be written in set-builder notation as «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»3«/mn»«mo»§#60;«/mo»«mi»x«/mi»«mo»§#60;«/mo»«mn»3«/mn»«mi mathvariant=¨normal¨»,«/mi»«mo»§#160;«/mo»«mi»x«/mi»«mo»§#8712;«/mo»«mi mathvariant=¨normal¨»R«/mi»«/mrow»«/mfenced»«/mstyle»«/math». It also can be written in interval notation as «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mfenced»«mrow»«mo»-«/mo»«mn»3«/mn»«mo»,«/mo»«mo»§#160;«/mo»«mn»3«/mn»«/mrow»«/mfenced»«/mstyle»«/math».
The interval represented above can be written in set-builder notation as «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»3«/mn»«mo»§#60;«/mo»«mi»x«/mi»«mo»§#60;«/mo»«mn»3«/mn»«mi mathvariant=¨normal¨»,«/mi»«mo»§#160;«/mo»«mi»x«/mi»«mo»§#8712;«/mo»«mi mathvariant=¨normal¨»R«/mi»«/mrow»«/mfenced»«/mstyle»«/math». It also can be written in interval notation as «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mfenced»«mrow»«mo»-«/mo»«mn»3«/mn»«mo»,«/mo»«mo»§#160;«/mo»«mn»3«/mn»«/mrow»«/mfenced»«/mstyle»«/math».
When both end points are included in the interval it is known as a closed interval. If the interval does not include the end points, it is called an open interval. Intervals that include only one end point are known as semi-open intervals.
Click the interactive button to experiment with the three different types of notations; inequalities, interval and set builder. You can explore how changing the points on a number line will change the notation. Note that in the animation, the end points are included at all times.
Interactive
Click the interactive button to experiment with the three different types of notations; inequalities, interval and set builder. You can explore how changing the points on a number line will change the notation. Note that in the animation, the end points are included at all times.
Skill Builder
For more information on number sets, click the Skill Builder button to access the Skill Builder page.