Investigating Inverses -...

Algebra 2 HS/Mathematics

Unit: 03 Lesson: 01

Suggested Duration: 5 days

Investigating Inverses

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Lesson Synopsis: Students explore inverses of relations by comparing graphs, tables, and algebraic generalizations. This is an introductory investigation that will be extended when comparing inverse relations, such as quadratic and square root function and exponential and logarithmic functions. TEKS:

2A.4 Algebra and geometry. The student connects algebraic and geometric representations of functions. The student is expected to:

2A.4C Describe and analyze the relationship between a function and its inverse.

GETTING READY FOR INSTRUCTION Performance Indicator(s):

Connect a function and its inverse by describing their graphs and analyzing respective tables of values. Explain the relationship between the domains and ranges of the function and its inverse. Compare algebraic generalizations of the function and its inverse. (2A.4C)

ELPS: 1E, 2E, 2I, 3D, 3G, 3H, 4F, 5G

Key Understandings and Guiding Questions: The graph of a function and its inverse reflect across the y = x line; the elements of the ordered pairs are

reversed; and the x and y variables are interchanged in the algebraic generalizations. — What observations can you make about the graph and its inverse? — What patterns do you observe in the tables of the relation and its inverse? — How are the domains and ranges of a relation and its inverse related? — How is this relationship reflected in the graph of the inverse? — What are the steps in solving a linear equation? — How are these steps similar to or different from finding the equation of its inverse?

Vocabulary of Instruction:

inverse relation inverse function Materials:

graphing calculator MIRA or other reflecting

drawing tool

colored pencils (2 colors per student)

straightedge

Resources:

STATE RESOURCES: — Mathematics TEKS Toolkit: Clarifying Activity/Lesson/Assessment

http://www.utdanacenter.org/mathtoolkit/index.php — MAP: Explore/Explain 1: Inverses of Functions – Part 1 (Generating an Inverse Function), Part 2 (Numerical

and Symbolic Representations), Part 3 (Investigating Linear Functions) http://www.tea.state.tx.us/math/index.html

— MAP: Student Lesson – Inverses of Functions http://www.tea.state.tx.us/math/index.html

— TEXTEAMS: Part I: Algebra II/Precalculus: I – Foundations of Functions; 3. Making Connections, 3.1 Inverses, Act. 1 (Parametric Equations), Act. 2 (Restricted Domains)


Unit: 03 Lesson: 01

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Advance Preparation: 1. Handout: Investigating Inverses (1 per student) 2. Handout: Inverse of a Line (1 per student) 3. Handout: Intriguing Inverses (1 per student) 4. Handout: Inverse of Lines, Again! (1 per student) 5. Handout: Inverses (1 per student)

Background Information:This unit introduces students to the concept of inverse relations. They investigate inverse relationships through graphing (reflecting a graph over y = x) and through tables. They also investigate interchanging the variables x and y in a linear equation and solving for y. In future units, as each function family is studied, the parent function, its family members and their inverses will be studied in depth. This unit is strictly introductory.

GETTING READY FOR INSTRUCTION SUPPLEMENTAL PLANNING DOCUMENT Instructors are encouraged to supplement, differentiate and substitute resources, materials, and activities to address the needs of learners. The Exemplar Lessons are one approach to teaching and reaching the Performance Indicators and Specificity in the Instructional Focus Document for this unit. A Microsoft Word template for this Planning document is located at www.cscope.us/sup_plan_temp.doc. If a supplement is created electronically, users are encouraged to upload the document to their Lesson Plans as a Lesson Plan Resource for future reference.

INSTRUCTIONAL PROCEDURES

Instructional Procedures Notes for Teacher

ENGAGE NOTE: 1 Day = 50 minutes Suggested Day 1

1. Distribute the handout: Investigating Inverses to each student. 2. Instruct students to graph the points in Table 1 in one color and connect

the points as they plot them. Do the same with the points in Table 2 with a different color pencil.

3. Have students work in pairs to answer the questions 1-3. 4. Conduct a whole class discussion in which students present their

findings. 5. Clarify any misconceptions—Students should recognize the graphs as

reflections of each other and the tables of values as having been interchanged.

MATERIALS Handout: Investigating Inverses (1

per student) colored pencils (2 colors per student) TEACHER NOTE The purpose of this activity is to set the stage for understanding inverses of relations. Do not introduce the term inverse yet. TEACHER NOTE Instruct students to keep this work. They will need it for the Explore activity.

STATE RESOURCES Mathematics TEKS Toolkit: Clarifying Activity/Lesson/Assessment may be used to reinforce these concepts or used as alternate activities. MAP: Explore/Explain 1: Inverses of Functions – Part 1 (Generating an Inverse Function), Part 2 (Numerical and Symbolic Representations), Part 3 (Investigating Linear Functions) may be used to reinforce these concepts or used as alternate activities. MAP: Student Lesson – Inverses of Functions may be used to reinforce


Unit: 03 Lesson: 01

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Instructional Procedures Notes for Teacher these concepts or used as alternate activities. TEXTEAMS: Part I: Algebra II/Precalculus I – Foundations of Functions; 3. Making Connections, 3.1 Inverses, Act. 1 (Parametric Equations), Act. 2 (Restricted Domains) may be used to reinforce these concepts or used as alternate activities.

EXPLORE/EXPLAIN 1 Suggested Day 2 1. Distribute the handout: Inverse of a Line to each student. 2. Have students work independently to complete questions 1-2. Ensure

each student has created his/her own linear equation by monitoring students.

3. Ask for volunteer responses for the answer to question 2 (linear equation). Ask students the following questions. Were the equations for each student the same? No, students

created their own equation to fit y = mx + b. What do all equations have in common? They all represent lines,

they all contain an x and y variable (unless someone used the horizontal, linear function).

What parts of the equations are different? The “m” and “b” values may differ from equation to equation.

What does the “m” value represent? The slope of the line What does the “b” value represent? The y-intercept of the line,

(0,b) 4. Have students work in pairs or small groups to complete questions 4-5. 5. Have students present their graphs and tables on chart paper or circulate

around the class to observe each others work. 6. After viewing the other work, have students complete question 6. 7. Refer students to their work from the Engage activity. Have students

continue to work in pairs or groups to complete questions 7-9. 8. Conduct a whole class discussion to share solutions. Clarify any

misconceptions. Students should understand that a graph and its reflection have interchanged x-values and y-values in their respective tables.

9. Introduce inverse. Explain that the pairs of graphs in both the Engage and in the Explore are inverses of each other. Ask students to describe inverse in their own words.

10. Elicit student definitions. Clarify any misconceptions.

MATERIALS Handout: Inverse of a Line (1 per

student) Student work from the Engage

activity colored pencils (2 colors per student) straightedge MIRA or other reflecting drawing

tool TEACHER NOTE In this activity, students will develop a deeper understanding of inverse relations by investigating inverse, linear relationships. TECHNOLOGY NOTE Students may share their graphs via the TI Navigator if technology is available. INFORMAL OBSERVATION Monitor students as they complete the assignment. Check for student understanding with the following questions. What is the same or different

about everyone’s graphs? What is the same or different

about everyone’s tables? Did it matter which equation you

started with?

MISCONCEPTION Students may have the misconception that the inverse of a function means to take the opposite sign or reciprocal of the function instead of switching the independent and dependent variables.


Unit: 03 Lesson: 01

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Instructional Procedures Notes for Teacher

STATE RESOURCES MAP: Explore/Explain 1: Inverses of Functions – Part 1 (Generating an Inverse Function), Part 2 (Numerical and Symbolic Representations), Part 3 (Investigating Linear Functions) may be used to reinforce these concepts or used as alternate activities. MAP: Student Lesson – Inverses of Functions may be used to reinforce these concepts or used as alternate activities.

ELABORATE Suggested Day 3 1. Distribute the handout: Intriguing Inverses to each student. 2. Have students work independently to complete questions 1-2. 3. Have students get into pairs or small groups to compare solutions. Share

out answers in whole group. Clarify any misunderstandings. 4. Have students continue working in pairs or small groups to complete

question 3. If necessary, this can be completed as homework.

MATERIALS Handout: Intriguing Inverses (1

copy per student) TEACHER NOTE In this activity students will extend their understanding of inverses of functions by comparing inverse relationships. TEACHER NOTE Question 2 is meant to assess student understanding of y = x as the underlying structure of the graph of a relation and its inverse. It is not meant to focus on names.

EXPLORE/EXPLAIN 2 Suggested Day 4 1. Debrief handout: Intriguing Inverses in whole group. Have students

correct the tables as needed. 2. Distribute the handout: Inverses of Lines, Again! to each student. 3. Have students work in pairs or small groups to complete questions 1-4 4. Stop students when finished with question 4. 5. Conduct a whole-class discussion. Elicit student responses for questions

2 and 3. Write their steps on the board overhead to compare. Clarify any misconceptions at this point. Ensure all students understand that the steps should be the same regardless of how they are verbalized.

6. Have students continue with questions 5-8. 7. Again, have students share and compare findings through a whole class

discussion. Clarify any misconceptions. See Teacher Note at right. In the original function, what is the input? What is the output?

The input is x and the output is y. If you solve an equation of the inverse, such as 8 = 2x + 6,

graphically what does the solution represent? The solution to the equation involving the inverse is the input (x) of the original function.

8. Have students work independently on question 9. If necessary, this can be completed as homework.

MATERIALS Handout: Inverses of Lines, Again!

(1 copy per student) graphing calculator TEACHER NOTE As function families are studied in future units, finding the equation of an inverse will be further emphasized. Do not expect mastery in this unit. The focus is on understanding that there is a connection between inverses and inverse operations. The original graphs the outputs versus the inputs. The inverse graphs the inputs versus the outputs.

STATE RESOURCES TEXTEAMS: Part I: Algebra II/Precalculus


Unit: 03 Lesson: 01

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Instructional Procedures Notes for Teacher I – Foundations of Functions; 3. Making Connections, 3.1 Inverses, Act. 1 (Parametric Equations), Act. 2 (Restricted Domains) may be used to reinforce these concepts or used as alternate activities.

EVALUATE Suggested Day 5 1. Debrief handout: Inverses of Lines, Again!, question 9 in whole group. 2. Distribute the handout: Inverses to each student. 3. Have students work individually on the handout: Inverses.

MATERIALS Handout: Inverses (1 per student) TEACHER NOTE Students should work on this activity independently. This activity should be used to assess student knowledge of the concepts covered in the lesson.


Unit: 03 Lesson: 01

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Investigating Inverses KEY

Table 1 Table 2

x y x y

5 3 3 5

2 7 7 2

0 1 1 0

2 4 4 2

4 4 4 4

4 6 6 4

5 7 7 5

1. Look at the ordered pairs in each table. What patterns do you observe?

The x-values and y-values switch.

2. For each table, plot the points in order, and connect them with line segments. To help tell the

two graphs apart, use a different color for each. 3. How are the two graphs related? What transformation occurs?

The graph of the points from Table 2 is the reflection of the graph of the points from Table 1.

4. A line of symmetry runs between these two graphs. Trace it with a dotted line. What is the equation for this line? The equation for the line is y = x.


Unit: 03 Lesson: 01

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Investigating Inverses

Table 1 Table 2

x y x y

5 3 3 5

2 7 7 2

0 1 1 0

2 4 4 2

4 4 4 4

4 6 6 4

5 7 7 5

1. Look at the ordered pairs in each table. What patterns do you observe?

2. For each table, plot the points in order, and connect them with line segments. To help tell the

two graphs apart, use a different color for each. 3. How are the two graphs related? What transformation occurs?

4. A line of symmetry runs between these two graphs. Trace it with a dotted line. What is the

equation for this line?


Unit: 03 Lesson: 01

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Inverse of a Line (pp. 1 of 2) KEY In this activity, you will continue to explore inverse relations by investigating inverses of linear functions. 1. Choose values for m and b to write a linear equation in slope-intercept form. Write the equation here: y = x + Student answers will vary.

2. Use your equation to complete the table for the “original line,” and plot the points to construct

the graph.

Original Line

Reflected Line

x y x y

3. Place the MIRA along the line y = x (dotted) and sketch the reflection of your line. As you do,

be sure to mark the image points reflected from those in your table. See student work.

4. Record the coordinates from your reflected line in the table on the right. Describe the patterns you observe.

Student answers will vary. Sample: The coordinates switch!

5. Is the reflected line a function? Why or why not? Yes, it is a function. There is one output for each input.


Unit: 03 Lesson: 01

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Inverse of a Line (pp. 2 of 2) KEY 6. Compare your graphs and tables with others in your class. What observations can you make? Students should observe that no matter the equation, everyone should have tables that show

interchanged x- and y-values.

7. Think back to the Engage activity (“Investigating Inverses”), shown at the right. How is the graph from the previous activity alike or different from the graphs in problems 1-4? In both, the graphs are reflections of each other over the line y = x.

8. How are the tables from the previous Engage activity like or different from the ones in problems 1-4?

In both sets of tables, the x- and y-values of the original are switched and become the y-and x-values of the reflected graph.


Unit: 03 Lesson: 01

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Inverse of a Line (pp. 1 of 2) In this activity, you will continue to explore inverse relations by investigating inverses of linear functions. 1. Choose values for m and b to write a linear equation in slope-intercept form. Write the equation here: y = x +

2. Use your equation to complete the table for the “original line,” and plot the points to construct

the graph.

Original Line

Reflected Line

x y x y

3. Place the MIRA along the line y = x (dotted) and sketch the reflection of your line. As you do,

be sure to mark the image points reflected from those in your table.

4. Record the coordinates from your reflected line in the table on the right. Describe the patterns you observe.

5. Is the reflected line a function? Why or why not? .


Unit: 03 Lesson: 01

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Inverse of a Line (pp. 2 of 2) 6. Compare your graphs and tables with others in your class. What observations can you make?

7. Think back to the Engage activity (“Investigating Inverses”), shown at the right. How is the graph from the previous activity alike or different from the graphs in problems 1-4?

8. How are the tables from the previous Engage activity like or different from the ones in problems 1-4?


Unit: 03 Lesson: 01

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Intriguing Inverses (pp. 1 of 3) KEY

1. List five ordered pairs from points on the function graphed below.

2. Use these points to name five ordered pairs that would be on the inverse relation of the function.

Function Inverse x y x y

Sam

ple

-6 5 5 -6

-5 0 0 -5

-3 4 4 -3

-1 0 0 -1

0 5 5 0

3. Is the inverse relation also a function? Why or why not?

No, the inverse relation is not a function. There is more than one output for some inputs.

4. Use a calculator to complete the table below for the function x

y1

.

Use these points to name ordered pairs that would be on the inverse relation of the function.

xy

1


-2 -0.5 -0.5 -2

-1 -1 -1 -1

-0.5 -2 -2 -0.5

-0.4 -2.5 -2.5 -0.4

0.4 2.5 2.5 0.4 0.5 2 2 0.5 1 1 1 1 2 0.5 0.5 2 5. What do you notice is special about the inverse relation of this function?

Student answers will vary. This function is its own inverse.


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Intriguing Inverses (pp. 2 of 3) KEY 6. For each of the given parent functions, create a table of values containing at least five points on

its graph. Use this table to create a table of values for its inverse relation. Graph the inverse relation of each parent function. Compare the characteristics of the parent function and its inverse relation.

Graph of Parent Function

Table of Points for

Parent Function

Table of Points for

Inverse

Graph of Inverse Relation

Comparison of Characteristics

Varies EX. The inverse relation is the same line. The parent function is the inverse of itself. Both are still functions.

Varies EX. The inverse relation of the parent function is not a function.

Varies EX. The inverse relation is the right side of y=x2. The inverse relation is a function.

Varies EX. The inverse relation of the parent function is a function.

x y -3 -3 -1 -1 0 0 2 2 5 5

x y -3 -3 -1 -1 0 0 2 2 5 5

x y -2 4 -1 1 0 0 1 1 2 4

x y 4 -2 1 -1 0 0 1 1 4 2

x y 0 0 1 1 4 2 7 2.6 9 3

x y 0 0 1 1 2 4

2.6 7 3 9

x y -1 1/2 0 1 1 2 2 4 3 8

x y 1/2 -1 1 0 2 1 4 2 8 3


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Intriguing Inverses (pp. 3 of 3) KEY


Table of Points for

Parent Function

Table of Points for

Inverse





Varies EX. The inverse graph is the same as the original graph. The inverse relation of the parent function is a function.

Varies EX. One side is on the y=x line and does not change. The other side reflects. The inverse relation of the parent function is not a function.

x y -2 4 -1 2 0 1 1 1/2 2 1/4

x y 4 -2 2 -1 1 0

1/2 1 1/4 2

x y 0.1 -1 0.5 -0.3 1 0 2 0.3

10 1

x y -1 0.1

-0.3 0.50 1

0.3 2 1 10

x y -3 -1/3 -1 -1

-1/3 -3 1/3 3 1 1 3 1/3

x y -1/3 -3 -1 -1 -3 -1/33 1/3 1 1

1/3 3

x y -6 6 -4 4 0 0 2 2 4 4 6 6

x y 6 -6 4 -4 0 0 2 2 4 4 6 6


Unit: 03 Lesson: 01

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Intriguing Inverses (pp. 1 of 3)

1. List five ordered pairs from points on the

function graphed below. 2. Use these points to name five ordered pairs

that would be on the inverse relation of the function.


3. Is the inverse relation also a function? Why or why not?

4. Use a calculator to complete the table below for the function x

y1

.

Use these points to name ordered pairs that would be on the inverse relation of the function.

xy

1


-2

-1

-0.5

-0.4

0.4 0.5 1 2 5. What do you notice is special about the inverse relation of this function?


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Intriguing Inverses (pp. 2 of 3) 6. For each of the given parent functions, create a table of values containing at least five points on

its graph. Use this table to create a table of values for its inverse relation. Graph the inverse relation of each parent function. Compare the characteristics of the parent function and its inverse relation.


Table of Points for

Parent Function

Table of Points for

Inverse



x y

x y

x y

x y

x y

x y

x y

x y


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Intriguing Inverses (pp. 3 of 3)


Table of Points for

Parent Function

Table of Points for

Inverse



x y

x y

x y

x y

x y

x y

x y

x y


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Inverses of Lines, Again! (pp. 1 of 2) KEY 1. Consider the linear equation 32

1 xy . Use this equation to complete the table.

x y

-4 1 2. In these cases, when starting with a y-value to find x, you can solve

an equation like the ones below. Record the necessary steps. -1 2.5

0 3 313 21 x

10 = ½ x 20 = x

347 21 x

44 = ½ x 88 = x

3 4.5

6 6

20 13 88 47 3. In the original function, imagine switching the x- and y-variables. If x and y are interchanged, the

new equation would be 321 yx . Solve this equation for y. Record your steps verbally.

y = 2x – 6. Subtract 3 from both sides. Multiply both sides by 2.

4. How are the processes used in questions 2 and 3 alike? The steps are the same.

5. Graph the original function 32

1 xy and the equation from question 3 ( 321 yx , solved for

y) on the same set of axes. 6. What do you notice about the graphs? They are reflections of each other about the

line y = x. (They are inverses.)

7. How are inverse relations connected to the inverse operations used to solve an equation?

Except for exchanging the x and y, the steps to find the solution to an equation are the same as finding the inverse of its line. The outputs of the original become the inputs of the inverse.


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Inverses of Lines, Again! (pp. 2 of 2) KEY

For the following equations, determine the inverse function, create a table for the original and inverse function, and graph the original and inverse function.

Original Function Inverse Function Tables Graph

y1 = 2x – 4 y2 = ½ x + 2

x y1 x y2 -3 -10 -10 -3 -2 -8 -8 -2 -1 -6 -6 -1 0 -4 -4 0 1 -2 -2 1 2 0 0 2 4 4 2 3 6 8 4 4

y1 = 131

x y2 = 3 x – 3

x y1 x y2 -9 -2 -2 -9 -6 -1 -1 -6 -3 0 0 -3 0 1 1 0 3 2 2 3 6 3 3 6 9 4 4 9


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Inverses of Lines, Again! (pp. 1 of 2) 1. Consider the linear equation 32

1 xy . Use this equation to complete the table.

x y

-4 2. In these cases, when starting with a y-value to find x, you can solve

an equation like the ones below. Record the necessary steps.

-1

0 313 21 x 347 2

1 x

3

6

13 47 3. In the original function, imagine switching the x- and y-variables. If x and y are interchanged, the

new equation would be 321 yx . Solve this equation for y. Record your steps verbally.

4. How are the processes used in questions 2 and 3 alike?

5. Graph the original function 32

1 xy and the equation from question 3 ( 321 yx , solved for

y) on the same set of axes. 6. What do you notice about the graphs?

7. How are inverse relations connected to the inverse operations used to solve an equation?


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Inverses of Lines, Again! (pp. 3 of 3)

1. For the following equations, determine the inverse function, create a table for the original and inverse function, and graph the original and inverse function.

Original Function Inverse Function Tables Graph

y = 2x – 4

x y1 x y2

y = 131

x

x y1 x y2


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Inverses (pp. 1 of 2) KEY 1. Graph the inverse relation of the function shown

below. 2. Explain the method that you used

to determine the inverse relation.

Sample answers: First graph the line of y = x.

Reflect the graph of the relation over this line.

List coordinates for points on the function. Switch the x and y values to plot points on the inverse relation.

Table 1 (Original) 3. Table 1 (on the left) gives points on a function. Fill in Table 2 with points on the inverse relation of the function.

Table 2 (Inverse) x y x y -5 6 6 -5 0 2 2 0 1 5 5 1 3 0 0 3

4. Explain how the domains and ranges of a relation and its inverse relation are related.

How is this shown in their tables? The domain of the relation becomes the range of the inverse. The range of the relation is the

domain of its inverse. This is shown in the tables when the x-values and y-values are interchanged.


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Inverses (pp. 2 of 2) KEY 5. Given the function y = 3x + 6, determine the inverse relation and verbally explain the process. x = 3y + 6 Switch the x and y variables.

x – 6 = 3y Solve for y. First subtract six from both sides. ½ x – 2 = y Next divide every term by 3. y = ½ x – 2 Inverse relation (is a function)

6. Consider the equation 43

2 xy .

A) Complete a table of values for this function, and use it to graph the equation. B) Create a table of values for the inverse relation of this function. Graph the inverse relation.

A) Original B) Inverse

x y x y

-6 -8 -8 -6

-3 -6 -6 -3

0 -4 -4 0

3 -2 -2 3

6 0 0 6

9 2 2 9

7. In slope-intercept form, what is the equation for the inverse relation? Is it a function?

Inverse relation: y = 3

2x + 6

It is a function


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Inverses (pp. 1 of 2) 1. Graph the inverse relation of the function shown

below. 2. Explain the method that you used

to determine the inverse relation.

Table 1 (Original) 3. Table 1 (on the left) gives points on a function. Fill in Table 2 with points on the inverse relation of the function.

Table 2 (Inverse) x y x y -5 6 0 2 1 5 3 0

4. Explain how the domains and ranges of a relation and its inverse relation are related.

How is this shown in their tables?


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Inverses (pp. 2 of 2) 5. Given the function y = 3x + 6, determine the inverse relation and verbally explain the process.

6. Consider the equation 43

2 xy .

A) Complete a table of values for this function, and use it to graph the equation. B) Create a table of values for the inverse relation of this function. Graph the inverse relation.

A) Original B) Inverse

x y x y

7. In slope-intercept form, what is the equation for the inverse relation? Is it a function?

Investigating Inverses -...

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