L1 Domain, Range, and Interval Notation - Practice 2
Completion requirements
Unit 1A
Precalculus
Lesson 1: Domain, Range, and Interval Notation
Practice
Once you feel confident with interval notation, click on the Practice tab and complete problems 1 to 8. Check your answers by going to the Solutions tab.
Instructions: Click the Download File button to download a printable PDF of the questions. Answer each of the following practice questions on a separate piece of paper. Step by step solutions are provided under the Solutions tab. You will learn the material more thoroughly if you complete the questions before checking the answers.
Instructions: Click the Download File button to download a printable PDF of the questions. Answer each of the following practice questions on a separate piece of paper. Step by step solutions are provided under the Solutions tab. You will learn the material more thoroughly if you complete the questions before checking the answers.
1.
In the previous question, the domain and range of «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»y«/mi»«mo»=«/mo»«mi»g«/mi»«mi mathvariant=¨normal¨»(«/mi»«mi»t«/mi»«mi mathvariant=¨normal¨»)«/mi»«/mrow»«/mstyle»«/math»
were expressed in set builder notation. Now write the function’s domain and range in interval notation.
2.
Express the interval represented by the graph below as an inequality and using interval notation. Is this an open, closed or semi-open interval? Explain.
3.
Express the interval represented by the graph below as an inequality, using interval notation and using set builder notation. Is this an open, closed or
semi-open interval? Explain.
4.
Sketch a graph of the following intervals and express each as an inequality.
a.
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mfenced»«mrow»«mo»-«/mo»«mn»2«/mn»«mo»,«/mo»«mo»§#160;«/mo»«mn»4«/mn»«/mrow»«/mfenced»«/mstyle»«/math»
b.
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mfenced open=¨[¨»«mrow»«mn»3«/mn»«mi mathvariant=¨normal¨»,«/mi»«mo»§#160;«/mo»«mn»7«/mn»«/mrow»«/mfenced»«/mstyle»«/math»
c.
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mfenced open=¨[¨ close=¨]¨»«mrow»«mo»§#8722;«/mo»«mfrac»«mn»1«/mn»«mn»2«/mn»«/mfrac»«mi mathvariant=¨normal¨»,«/mi»«mo»§#160;«/mo»«mo»§#960;«/mo»«/mrow»«/mfenced»«/mstyle»«/math»
5.
Using the graph below write the corresponding inequality and express the solution by using interval notation.
6.
Sketch a graph of the solution set represented by «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»«mi»x«/mi»«mo»§#62;«/mo»«mn»2«/mn»«mi
mathvariant=¨normal¨»,«/mi»«mo»§#160;«/mo»«mi»x«/mi»«mo»§#8712;«/mo»«mi mathvariant=¨normal¨»R«/mi»«/mrow»«/mfenced»«/mstyle»«/math». Express the set using interval notation.
7.
Using interval notation, how could you express all real numbers?(Hint: This set includes all numbers from negative infinity to positive infinity.)
8.
Represent the inequality «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»x«/mi»«mo»§#8805;«/mo»«mn»6«/mn»«/mrow»«/mstyle»«/math» with a number line and using interval notation.
1.
In the previous question, the domain and range of «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»y«/mi»«mo»=«/mo»«mi»g«/mi»«mi mathvariant=¨normal¨»(«/mi»«mi»t«/mi»«mi
mathvariant=¨normal¨»)«/mi»«/mrow»«/mstyle»«/math» were expressed in set builder notation. Now write the function’s domain and range in interval notation.
Remember «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mo»§#8746;«/mo»«/mstyle»«/math» represents the union of two sets. It means both sets are part of the domain/range.
Domain: «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mfenced»«mrow»«mo»§#8722;«/mo»«mo»§#8734;«/mo»«mi mathvariant=¨normal¨»,«/mi»«mo»§#160;«/mo»«mn»2«/mn»«/mrow»«/mfenced»«mo»§#8746;«/mo»«mfenced»«mrow»«mn»2«/mn»«mi mathvariant=¨normal¨»,«/mi»«mo»§#160;«/mo»«mo»§#8734;«/mo»«/mrow»«/mfenced»«/mstyle»«/math»
Range: «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mfenced close=¨]¨»«mrow»«mo»§#8722;«/mo»«mo»§#8734;«/mo»«mo»,«/mo»«mo»§#160;«/mo»«mn»2«/mn»«/mrow»«/mfenced»«mo»§#8746;«/mo»«mo»[«/mo»«mn»6«/mn»«mo»,«/mo»«mo»§#160;«/mo»«mo»§#8734;«/mo»«mo»)«/mo»«/mstyle»«/math»
Domain: «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mfenced»«mrow»«mo»§#8722;«/mo»«mo»§#8734;«/mo»«mi mathvariant=¨normal¨»,«/mi»«mo»§#160;«/mo»«mn»2«/mn»«/mrow»«/mfenced»«mo»§#8746;«/mo»«mfenced»«mrow»«mn»2«/mn»«mi mathvariant=¨normal¨»,«/mi»«mo»§#160;«/mo»«mo»§#8734;«/mo»«/mrow»«/mfenced»«/mstyle»«/math»
Range: «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mfenced close=¨]¨»«mrow»«mo»§#8722;«/mo»«mo»§#8734;«/mo»«mo»,«/mo»«mo»§#160;«/mo»«mn»2«/mn»«/mrow»«/mfenced»«mo»§#8746;«/mo»«mo»[«/mo»«mn»6«/mn»«mo»,«/mo»«mo»§#160;«/mo»«mo»§#8734;«/mo»«mo»)«/mo»«/mstyle»«/math»
2.
Express the interval represented by the graph below as an inequality and using interval notation. Is this an open, closed or semi-open interval? Explain.
Inequality: «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mo»§#8722;«/mo»«mn»1«/mn»«mo»§#60;«/mo»«mi»x«/mi»«mo»§#60;«/mo»«mn»4«/mn»«/mrow»«/mstyle»«/math»
Interval Notation: «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»«mn»4«/mn»«/mrow»«/mfenced»«/mstyle»«/math»
It is an open Interval because the end points are not included.
Interval Notation: «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»«mn»4«/mn»«/mrow»«/mfenced»«/mstyle»«/math»
It is an open Interval because the end points are not included.
3.
Express the interval represented by the graph below as an inequality, using interval notation and using set builder notation. Is this an open, closed or semi-open interval? Explain.
Inequality: «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mo»§#8722;«/mo»«mn»2«/mn»«mo»§#60;«/mo»«mi»x«/mi»«mo»§#8804;«/mo»«mn»5«/mn»«/mrow»«/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»5«/mn»«/mrow»«/mfenced»«/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»5«/mn»«mi mathvariant=¨normal¨»,«/mi»«mo»§#160;«/mo»«mi»x«/mi»«mo»§#8712;«/mo»«mi mathvariant=¨normal¨»R«/mi»«/mrow»«/mfenced»«/mstyle»«/math»
This is a semi-open interval since only one end point is included.
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»5«/mn»«/mrow»«/mfenced»«/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»5«/mn»«mi mathvariant=¨normal¨»,«/mi»«mo»§#160;«/mo»«mi»x«/mi»«mo»§#8712;«/mo»«mi mathvariant=¨normal¨»R«/mi»«/mrow»«/mfenced»«/mstyle»«/math»
This is a semi-open interval since only one end point is included.
4.
Sketch a graph of the following intervals and express each as an inequality.
a.
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mfenced»«mrow»«mo»-«/mo»«mn»2«/mn»«mo»,«/mo»«mo»§#160;«/mo»«mn»4«/mn»«/mrow»«/mfenced»«/mstyle»«/math»
b.
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mfenced open=¨[¨»«mrow»«mn»3«/mn»«mi mathvariant=¨normal¨»,«/mi»«mo»§#160;«/mo»«mn»7«/mn»«/mrow»«/mfenced»«/mstyle»«/math»
c.
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mfenced open=¨[¨ close=¨]¨»«mrow»«mo»§#8722;«/mo»«mfrac»«mn»1«/mn»«mn»2«/mn»«/mfrac»«mi mathvariant=¨normal¨»,«/mi»«mo»§#160;«/mo»«mo»§#960;«/mo»«/mrow»«/mfenced»«/mstyle»«/math»
Graph
Inequality
a.
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mo»§#8722;«/mo»«mn»2«/mn»«mo»§#60;«/mo»«mi»x«/mi»«mo»§#60;«/mo»«mn»4«/mn»«/mrow»«/mstyle»«/math»
b.
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mn»3«/mn»«mo»§#8804;«/mo»«mi»x«/mi»«mo»§#60;«/mo»«mn»7«/mn»«/mrow»«/mstyle»«/math»
c.
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mo»§#8722;«/mo»«mfrac»«mn»1«/mn»«mn»2«/mn»«/mfrac»«mo»§#8804;«/mo»«mi»x«/mi»«mo»§#8804;«/mo»«mo»§#960;«/mo»«/mrow»«/mstyle»«/math»
5.
Using the graph below write the corresponding inequality and express the solution by using interval notation.
Inequality: «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»x«/mi»«mo»§#8804;«/mo»«mn»3«/mn»«/mrow»«/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»«mo»§#8734;«/mo»«mi mathvariant=¨normal¨»,«/mi»«mo»§#160;«/mo»«mn»3«/mn»«/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»«mo»§#8734;«/mo»«mi mathvariant=¨normal¨»,«/mi»«mo»§#160;«/mo»«mn»3«/mn»«/mrow»«/mfenced»«/mstyle»«/math»
6.
Sketch a graph of the solution set represented by «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»«mi»x«/mi»«mo»§#62;«/mo»«mn»2«/mn»«mi
mathvariant=¨normal¨»,«/mi»«mo»§#160;«/mo»«mi»x«/mi»«mo»§#8712;«/mo»«mi mathvariant=¨normal¨»R«/mi»«/mrow»«/mfenced»«/mstyle»«/math». Express the set using interval notation.
Graph
Interval Notation: «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mfenced»«mrow»«mn»2«/mn»«mi mathvariant=¨normal¨»,«/mi»«mo»§#160;«/mo»«mo»§#8734;«/mo»«/mrow»«/mfenced»«/mstyle»«/math»
Interval Notation: «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mfenced»«mrow»«mn»2«/mn»«mi mathvariant=¨normal¨»,«/mi»«mo»§#160;«/mo»«mo»§#8734;«/mo»«/mrow»«/mfenced»«/mstyle»«/math»
7.
Using interval notation, how could you express all real numbers? (Hint: This set includes all numbers from negative infinity to positive infinity.)
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mfenced»«mrow»«mo»§#8722;«/mo»«mo»§#8734;«/mo»«mi mathvariant=¨normal¨»,«/mi»«mo»§#160;«/mo»«mo»§#8734;«/mo»«/mrow»«/mfenced»«/mstyle»«/math»
8.
Represent the inequality «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»x«/mi»«mo»§#8805;«/mo»«mn»6«/mn»«/mrow»«/mstyle»«/math» with a number line and using interval notation.
Graph
Interval Notation: «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mfenced open=¨[¨»«mrow»«mn»6«/mn»«mi mathvariant=¨normal¨»,«/mi»«mo»§#160;«/mo»«mo»§#8734;«/mo»«/mrow»«/mfenced»«/mstyle»«/math»
Interval Notation: «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mfenced open=¨[¨»«mrow»«mn»6«/mn»«mi mathvariant=¨normal¨»,«/mi»«mo»§#160;«/mo»«mo»§#8734;«/mo»«/mrow»«/mfenced»«/mstyle»«/math»