L1 Domain, Range, and Interval Notation - Part 3
Completion requirements
Unit 1A
Precalculus
Lesson 1: Domain, Range, and Interval Notation
Determine the domain and range of the relation shown using set builder notation.
In words, the domain is the set of all «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»x«/mi»«/mstyle»«/math» such that «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle
mathsize=¨14px¨»«mi»x«/mi»«/mstyle»«/math» is less than four, and «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»x«/mi»«/mstyle»«/math» is a Real Number. In set builder notation,
the domain is written as «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mfenced open=¨{¨ close=¨}¨»«mrow»«mi»x«/mi»«mo»§#8739;«/mo»«mi mathvariant=¨normal¨»x«/mi»«mo»§#60;«/mo»«mn»4«/mn»«mo»,«/mo»«mo»§#160;«/mo»«mi
mathvariant=¨normal¨»x«/mi»«mo»§#8712;«/mo»«mi mathvariant=¨normal¨»R«/mi»«/mrow»«/mfenced»«/mstyle»«/math».
The range, in words, is the set of all «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»y«/mi»«/mstyle»«/math» such that «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»y«/mi»«/mstyle»«/math» is greater than negative five, and «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»y«/mi»«/mstyle»«/math» is a Real Number. In set builder notation, the range is written as «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mfenced open=¨{¨ close=¨}¨»«mrow»«mi»y«/mi»«mo»§#8739;«/mo»«mi»y«/mi»«mo»§#62;«/mo»«mo»-«/mo»«mn»5«/mn»«mo»,«/mo»«mo»§#160;«/mo»«mi»y«/mi»«mo»§#8712;«/mo»«mi mathvariant=¨normal¨»R«/mi»«/mrow»«/mfenced»«/mstyle»«/math».
The range, in words, is the set of all «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»y«/mi»«/mstyle»«/math» such that «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»y«/mi»«/mstyle»«/math» is greater than negative five, and «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»y«/mi»«/mstyle»«/math» is a Real Number. In set builder notation, the range is written as «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mfenced open=¨{¨ close=¨}¨»«mrow»«mi»y«/mi»«mo»§#8739;«/mo»«mi»y«/mi»«mo»§#62;«/mo»«mo»-«/mo»«mn»5«/mn»«mo»,«/mo»«mo»§#160;«/mo»«mi»y«/mi»«mo»§#8712;«/mo»«mi mathvariant=¨normal¨»R«/mi»«/mrow»«/mfenced»«/mstyle»«/math».
For examples demonstrating how to determine the domain and range of a relation, watch the video, Domain and Range, shown below.
Note: In this video, the narrator uses the phrase stick in. The proper terminology for this is substitute in. Please give the video time to load.
As seen in the video, Domain and Range, a function’s range can be determined algebraically. However, for this example, the graph of the function will be used to determine its range.
The graph of «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»f«/mi»«mfenced»«mi»x«/mi»«/mfenced»«/mrow»«/mstyle»«/math» is shown below.
As seen in the graph, the «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»y«/mi»«/mstyle»«/math»-values can be both very large and very small numbers. The only value «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»y«/mi»«/mstyle»«/math» cannot be is «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»1«/mn»«/mstyle»«/math». Therefore, the range is «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mfenced open=¨{¨ close=¨}¨»«mrow»«mi»y«/mi»«mo»§#8739;«/mo»«mi»y«/mi»«mo»§#8800;«/mo»«mn»1«/mn»«mo»,«/mo»«mo»§#160;«/mo»«mi»y«/mi»«mo»§#8712;«/mo»«mi mathvariant=¨normal¨»R«/mi»«/mrow»«/mfenced»«/mstyle»«/math».
Note: In this video, the narrator uses the phrase stick in. The proper terminology for this is substitute in. Please give the video time to load.
Determine the domain and range of the function «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»f«/mi»«mfenced»«mi»x«/mi»«/mfenced»«mo»=«/mo»«mfrac»«mi»x«/mi»«mrow»«mi»x«/mi»«mo»-«/mo»«mn»2«/mn»«/mrow»«/mfrac»«/mrow»«/mstyle»«/math»
using set builder notation.
For the domain, start by finding the restrictions on «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»x«/mi»«/mstyle»«/math». The restrictions state the values of «math style=¨font-family:Verdana¨
xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»x«/mi»«/mstyle»«/math» that make the function invalid. For «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»f«/mi»«mfenced»«mi»x«/mi»«/mfenced»«mo»=«/mo»«mfrac»«mi»x«/mi»«mrow»«mi»x«/mi»«mo»-«/mo»«mn»2«/mn»«/mrow»«/mfrac»«/mrow»«/mstyle»«/math»,
the function is undefined, when the denominator, of «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»x«/mi»«mo»-«/mo»«mn»2«/mn»«/mrow»«/mstyle»«/math», is equal to zero.
This means the restriction on «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»x«/mi»«/mstyle»«/math» for, «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»f«/mi»«mfenced»«mi»x«/mi»«/mfenced»«mo»=«/mo»«mfrac»«mi»x«/mi»«mrow»«mi»x«/mi»«mo»-«/mo»«mn»2«/mn»«/mrow»«/mfrac»«/mrow»«/mstyle»«/math», is «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»x«/mi»«mo»§#8800;«/mo»«mn»2«/mn»«/mrow»«/mstyle»«/math». Written in set builder notation, the domain is expressed as «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mfenced open=¨{¨ close=¨}¨»«mrow»«mi»x«/mi»«mo»§#8739;«/mo»«mi mathvariant=¨normal¨»x«/mi»«mo»§#8800;«/mo»«mn»2«/mn»«mo»,«/mo»«mo»§#160;«/mo»«mi mathvariant=¨normal¨»x«/mi»«mo»§#8712;«/mo»«mi mathvariant=¨normal¨»R«/mi»«/mrow»«/mfenced»«/mstyle»«/math».
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mtable columnalign=¨right center left¨»«mtr»«mtd»«mi»x«/mi»«mo»-«/mo»«mn»2«/mn»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»0«/mn»«/mtd»«/mtr»«mtr»«mtd»«mi»x«/mi»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»2«/mn»«/mtd»«/mtr»«/mtable»«/mstyle»«/math»
This means the restriction on «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»x«/mi»«/mstyle»«/math» for, «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»f«/mi»«mfenced»«mi»x«/mi»«/mfenced»«mo»=«/mo»«mfrac»«mi»x«/mi»«mrow»«mi»x«/mi»«mo»-«/mo»«mn»2«/mn»«/mrow»«/mfrac»«/mrow»«/mstyle»«/math», is «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»x«/mi»«mo»§#8800;«/mo»«mn»2«/mn»«/mrow»«/mstyle»«/math». Written in set builder notation, the domain is expressed as «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mfenced open=¨{¨ close=¨}¨»«mrow»«mi»x«/mi»«mo»§#8739;«/mo»«mi mathvariant=¨normal¨»x«/mi»«mo»§#8800;«/mo»«mn»2«/mn»«mo»,«/mo»«mo»§#160;«/mo»«mi mathvariant=¨normal¨»x«/mi»«mo»§#8712;«/mo»«mi mathvariant=¨normal¨»R«/mi»«/mrow»«/mfenced»«/mstyle»«/math».
The graph of «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»f«/mi»«mfenced»«mi»x«/mi»«/mfenced»«/mrow»«/mstyle»«/math» is shown below.
As seen in the graph, the «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»y«/mi»«/mstyle»«/math»-values can be both very large and very small numbers. The only value «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»y«/mi»«/mstyle»«/math» cannot be is «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»1«/mn»«/mstyle»«/math». Therefore, the range is «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mfenced open=¨{¨ close=¨}¨»«mrow»«mi»y«/mi»«mo»§#8739;«/mo»«mi»y«/mi»«mo»§#8800;«/mo»«mn»1«/mn»«mo»,«/mo»«mo»§#160;«/mo»«mi»y«/mi»«mo»§#8712;«/mo»«mi mathvariant=¨normal¨»R«/mi»«/mrow»«/mfenced»«/mstyle»«/math».
State the domain and range of «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»g«/mi»«mfenced»«mi»x«/mi»«/mfenced»«mo»=«/mo»«mfrac»«msqrt»«mi»x«/mi»«mo»+«/mo»«mn»4«/mn»«/msqrt»«mrow»«mi»x«/mi»«mo»+«/mo»«mn»3«/mn»«/mrow»«/mfrac»«/mrow»«/mstyle»«/math».
Graph the function.
The restriction on «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»x«/mi»«/mstyle»«/math» can be found by setting the denominator equal to zero.
Therefore, the restriction on «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»x«/mi»«/mstyle»«/math» is «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»x«/mi»«mo»§#8800;«/mo»«mo»-«/mo»«mn»3«/mn»«/mrow»«/mstyle»«/math».
In addition, the numerator of «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»g«/mi»«mfenced»«mi»x«/mi»«/mfenced»«mo»=«/mo»«mfrac»«msqrt»«mi»x«/mi»«mo»+«/mo»«mn»4«/mn»«/msqrt»«mrow»«mi»x«/mi»«mo»+«/mo»«mn»3«/mn»«/mrow»«/mfrac»«/mrow»«/mstyle»«/math» has generates an additional restriction on «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»x«/mi»«/mstyle»«/math». As the numerator contains a radical, «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»x«/mi»«mo»+«/mo»«mn»4«/mn»«mo»§#8805;«/mo»«mn»0«/mn»«/mrow»«/mstyle»«/math». As such, «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»x«/mi»«mo»§#8805;«/mo»«mo»-«/mo»«mn»4«/mn»«/mstyle»«/math».
Combining the two restrictions on «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»x«/mi»«/mstyle»«/math», the domain is «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mfenced open=¨{¨ close=¨}¨»«mrow»«mi»x«/mi»«mo»§#8739;«/mo»«mi mathvariant=¨normal¨»x«/mi»«mo»§#8805;«/mo»«mo»-«/mo»«mn»4«/mn»«mo»,«/mo»«mo»§#160;«/mo»«mi»x«/mi»«mo»§#8800;«/mo»«mo»-«/mo»«mn»3«/mn»«mo»,«/mo»«mo»§#160;«/mo»«mi mathvariant=¨normal¨»x«/mi»«mo»§#8712;«/mo»«mi mathvariant=¨normal¨»R«/mi»«/mrow»«/mfenced»«/mstyle»«/math».
Graph the function to determine the range.
From the graph, all values of «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»y«/mi»«/mstyle»«/math» are present. Therefore, the range is «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mfenced open=¨{¨ close=¨}¨»«mrow»«mi»y«/mi»«mo»§#8739;«/mo»«mi»y«/mi»«mo»§#8712;«/mo»«mi mathvariant=¨normal¨»R«/mi»«/mrow»«/mfenced»«/mstyle»«/math».
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mtable columnalign=¨right center left¨»«mtr»«mtd»«mi»x«/mi»«mo»+«/mo»«mn»3«/mn»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»0«/mn»«/mtd»«/mtr»«mtr»«mtd»«mi»x«/mi»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mo»-«/mo»«mn»3«/mn»«/mtd»«/mtr»«/mtable»«/mstyle»«/math»
Therefore, the restriction on «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»x«/mi»«/mstyle»«/math» is «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»x«/mi»«mo»§#8800;«/mo»«mo»-«/mo»«mn»3«/mn»«/mrow»«/mstyle»«/math».
In addition, the numerator of «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»g«/mi»«mfenced»«mi»x«/mi»«/mfenced»«mo»=«/mo»«mfrac»«msqrt»«mi»x«/mi»«mo»+«/mo»«mn»4«/mn»«/msqrt»«mrow»«mi»x«/mi»«mo»+«/mo»«mn»3«/mn»«/mrow»«/mfrac»«/mrow»«/mstyle»«/math» has generates an additional restriction on «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»x«/mi»«/mstyle»«/math». As the numerator contains a radical, «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»x«/mi»«mo»+«/mo»«mn»4«/mn»«mo»§#8805;«/mo»«mn»0«/mn»«/mrow»«/mstyle»«/math». As such, «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»x«/mi»«mo»§#8805;«/mo»«mo»-«/mo»«mn»4«/mn»«/mstyle»«/math».
Combining the two restrictions on «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»x«/mi»«/mstyle»«/math», the domain is «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mfenced open=¨{¨ close=¨}¨»«mrow»«mi»x«/mi»«mo»§#8739;«/mo»«mi mathvariant=¨normal¨»x«/mi»«mo»§#8805;«/mo»«mo»-«/mo»«mn»4«/mn»«mo»,«/mo»«mo»§#160;«/mo»«mi»x«/mi»«mo»§#8800;«/mo»«mo»-«/mo»«mn»3«/mn»«mo»,«/mo»«mo»§#160;«/mo»«mi mathvariant=¨normal¨»x«/mi»«mo»§#8712;«/mo»«mi mathvariant=¨normal¨»R«/mi»«/mrow»«/mfenced»«/mstyle»«/math».
Graph the function to determine the range.
From the graph, all values of «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»y«/mi»«/mstyle»«/math» are present. Therefore, the range is «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mfenced open=¨{¨ close=¨}¨»«mrow»«mi»y«/mi»«mo»§#8739;«/mo»«mi»y«/mi»«mo»§#8712;«/mo»«mi mathvariant=¨normal¨»R«/mi»«/mrow»«/mfenced»«/mstyle»«/math».
Skill Builder
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