TASK 1.4.1: INVERSES RELATIONS AND FUNCTIONS 3 3 2 Function 4 Symbolic Representation: ... In Exercise 3, ... that the vertical line test helps determine if a relation is a function

Algebra II: Strand 1. Foundations of Functions; Topic 4. Making Connections; Task 1.4.1

December 10, 2004. Ensuring Teacher Quality: Algebra II, produced by the Charles A. Dana Center at The University of Texas at Austin for the Texas Higher Education Coordinating Board.

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TASK 1.4.1: INVERSES, RELATIONS, AND FUNCTIONS Solutions Part I: On the following chart, give an example of a function for each representation listed. There are multiple solutions. The following are examples of 4 functions represented in different ways.

Function 1 Verbal Representation: The candy bar that I get from the vending machine depends upon the code that I enter when prompted.

Function 2 Graphical Representation:

Function 3 Tabular Representation:

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

Function 4 Symbolic Representation: y=x2



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Part II: 1. For the graphs of each of the relations given, list what kinds of restrictions we would have to

place on the x’s or y’s or both so that the relations will be functions of x. Fill in the given table with at least two ways (if possible) to restrict the x’s or y’s or both so that the result will give a function of x. There are multiple solutions to these representations (see the worksheet). See Exercise 3 in part II for possible solutions.

2. The graphs of each of the relations from Exercise 1 are given below. For each, sketch the

inverse relation using patty paper as a guide. Which (if any) of these inverse relations are functions? In Exercise 3, part II the graphs of the inverse relations are provided. Note that only those inverse relations that are functions are investigated in exercise 3. Note that in some of these exercises restricting the x-values imposes a non-ambiguous restriction on the y-values (so, the restriction would not necessarily need to be listed) or vice versa. However, we ask for both restrictions here so that in the exercises that follow participants can easily compare their findings and conclusions about domain and range.

3. In the table below, the graphs of relations a, b, c, e, and f from Exercise 1 are given along

with their respective inverse relations. The original relation is given as a solid curve and its inverse relation as a dotted curve. Also, portions of the original relation are shaded to indicate those portions that, under appropriate restrictions of x’s and y’s, are functions. Fill in the given table. Explain the relationship between the domain and range of a given function and the domain and range of its inverse function. Include an example in your explanation.

Graphs

Relation—solid graph Inverse Relation—dotted graph

Restriction of relation on

x-axis


y-axis

Domain of inverse function

Range of inverse function

a) Indicated by darker outline: x>1

Indicated by darker outline: y>0

Indicated by darker (dotted) outline: x>0

Indicated by darker (dotted) outline: y>1



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Graphs Relation—solid graph

Inverse Relation—dotted graph


x-axis


y-axis



Indicated by lighter outline: x>1

Indicated by lighter outline: y<0

Indicated by lighter (dotted) outline: x<0

Indicated by lighter (dotted) outline: y>1

b)

Indicated by darker outline: None !" < x < "

Indicated by darker outline: None !" < y < "

Indicated by darker (dotted) outline: None !" < x < "

Indicated by darker (dotted) outline: None !" < y < "

c) Indicated by darker outline: None !" < x < "

Indicated by darker outline:

!1.6 < y < 1.6

Indicated by darker (dotted) outline:

!1.6 < x < 1.6

Indicated by darker (dotted) outline: None !" < y < "



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x-axis


y-axis



Indicated by lighter outline: None !" < x < "

Indicated by lighter outline:

!4.7 < y < !1.6

Indicated by lighter (dotted) outline: !4.7 < x < !1.6

Indicated by lighter (dotted) outline: None !" < y < "

Indicated by lightest outline: None !" < x < "

Indicated by lightest outline:

1.6 < y < 4.7

Indicated by lightest(dotted) outline: 1.6 < x < 4.7

Indicated by lightest(dotted) outline: None !" < y < "

Indicated by darker outline: x>-2

Indicated by darker outline: y>1.5

Indicated by darker (dotted) outline: x>1.5

Indicated by darker (dotted) outline: y>-2

Indicated by lighter outline: x<1/2

Indicated by lighter outline: y<-1/2

Indicated by lighter (dotted) outline: x<-1/2

Indicated by lighter (dotted) outline: y<1/2

e)

Indicated by lightest outline: -2<x<1/2

Indicated by lightest outline: -1/2<y<3/2

Indicated by lightest (dotted) outline: -1/2<x<3/2

Indicated by lightest (dotted) outline: -2<y<1/2



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x-axis


y-axis



Indicated by darker outline: x>1

Indicated by darker outline: y>0

Indicated by darker (dotted) outline: x>0

Indicated by darker (dotted) outline: y>1

f)

Indicated by lighter outline: x>1

Indicated by lighter outline: y<0

Indicated by lighter (dotted) outline: x<0

Indicated by lighter (dotted) outline y>1

4. Explain the “vertical line test” and the “horizontal line test” that are used when referring to

functions. Use the graphs in Exercise 3 in your explanation. Participants should notice from their graphs and the discussion about relations and functions that the vertical line test helps determine if a relation is a function of x because it is a quick way of examining whether or not there is more than one y-value corresponding to a single x-value. Participants should observe from the graphs that the horizontal line test is a quick way to determine if there is more than one x-value for a given y-value. This helps determine whether or not the inverse of the function (as a relation) will be an inverse function.

Math notes Leaders must emphasize the relationship between the domain and range of a function and the domain and range of its inverse function.

Teaching notes

Because of the qualitative nature of this task, we have provided five instructor transparencies for some instruction to take place upfront after Part I. In Part I, the goal is to get an idea of the participants’ current knowledge and/or intuition about functions. Once participants have had a chance to work on this individually and in groups, hand out a blank transparency to each of the groups, have them write up their descriptions, and then present to the class. Some participants may ask if they need to give a definition of function for “verbal representation.” If this question arises, let the participants know that a formal definition will be discussed later and that the verbal representation should be that of a situation in which a function relationship is implied.



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Some terminology, including relations, inverse relations, functions, and inverse functions, needs to be discussed before participants continue with Part II of Task 1.4.1 (see transparencies provided at the end of this section). On instructor transparency 3, a graph is provided that shows the relation, the inverse relation, the line y=x, and a dotted square. This square it provided to emphasize the symmetry about the line y=x of a relation and its inverse relation. The bottom of transparency 4 is to be discussed lightly and then brought up again when participants are discussing their explanations to Exercise 3 in Part II (where they have to explain what the relationship is between domain and range of a function and domain and range of its inverse function). It is important that Exercise 3 of Part II is not handed out to the participants until they have had a chance to work on Exercises 1 and 2. It is also important in Exercise 1 of Part II that two transparencies of each page of Exercise 1 are distributed among the groups so that participants can record their findings. This is an excellent activity for bringing out multiple solutions and visualizations among the students. In Exercise 2, some guidance on how to use the patty paper to draw the inverses is necessary. After all participants in a group have worked Exercises 1 and 2, distribute Exercise 3 (& 4) to the group. They can compare the graphs from Exercise 3 with their results in Exercise 1. Instructor transparency 5 is a task to model for participants before they begin Exercise 3; a possible solution follows the transparency page. Exercise 4 may be assigned as homework. At the beginning of class the following day, ask the participants (in groups of 4) to compare their answers as they pass their papers in a clockwise fashion until all group members have read the answers proposed by the other group members. Ask each group to generate a group answer on chart paper. After all groups have posted their answers, ask each representative from each group to come forward to explain their group answer.

Technology notes On instructor transparency 3, graphs are provided for a relation, its inverse relation, etc. Participants can be shown how to enter lists into their calculator and use stat plot to plot points (L1, L2) and (L2, L1). For example: Go to STAT select Edit. Set L1=seq(N, N, -10, 10, .5) and L2=L1^2. This will give you two lists. Here is a sample screen shot of the lists:

Go to STAT PLOT and turn on Plots 1 and 2.



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Set Plot 1 so that the points (L1, L2) will be plotted. Set Plot 2 so that the points (L2, L1) will be plotted. Choose a friendly window and display your plots.



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Instructor Transparency 1 A relation is a correspondence between two sets, say A and B. If a is an element of A and b is an element of B and if a relation exists between a and b, then we say that a corresponds to b. We may write a corresponds to b as the ordered pair (a, b). Some examples: • Correspondence between each student in a school and their weight

in pounds. The relation may be called “weighs.” An example of an ordered pair for this relation would be (John, 156) which means John “weighs” 156 lbs.

• Correspondence between a person’s homepage and a page to

which it is linked. The relation may be called “links to.” An example of an ordered pair for this relation would be (James’s homepage, UT Austin’s web page) which means James’s homepage “links to” UT Austin’s web page.

• Correspondence between a woman and her child. The relation may be called “mother of.” An example of an ordered pair for this relation would be (Olga, Jonas) which means that Olga is the “mother of” Jonas.

• {(-4, 5), (-3, -4), (-3, -3), (-2, 0), (1, 1), (1, 2), (2, -4)}



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Instructor Transparency 2

Inverse Relations: A pair of relations are inverse relations if and only if whenever one relation contains the element (a, b), the other relation contains the element (b, a). Example: If we begin with the relation {(-4, 5), (-3, -4), (-3, -3), (-2, 0), (1, 1), (1, 2), (2, -4)} and want to create the inverse relation, we interchange the first and second components of the ordered pairs to get {(5, -4), (-4, -3), (-3, -3), (0, -2), (1, 1), (2, 1), (-4, 2)}. In tabular format, we interchange the left and right columns.

Relation 1 Inverse of Relation 1

x y x y -4 5 5 -4 -3 -4 -4 -3 -3 -3 -3 -3 -2 0 0 -2 1 1 1 1 1 2 2 1 2 -4 -4 2



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Graphically we see:



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Instructor Transparency 4 A function is a relation that associates with each element from a set A exactly one element of a set B. Some examples: • Correspondence between persons in the USA with social security

numbers and social security numbers. The function may be called “social security number.”

• Correspondence between students in a classroom and their

mothers. The function may be called “child of.” • Correspondence between a number and its square. The function

may be called “squared.” • {(-4, 5), (-3, -4), (-2, 1), (0, 0), (2, 1), (3, 4), (4, 5)}

For a function f, we can create an inverse relation of the function by taking all ordered pairs (a, f(a)) and plotting (f(a), a). This relation is called the inverse function if and only if the new relation (whose ordered pairs are given by (f(a), a)) is a function itself. We denote the inverse function of f by

f!1. Ordered pairs for

f!1 can also be written

b, f !1(b)( ) . It follows that if b = f (a) then f !1(b) = f !1( f (a)) = a .

Thus, we have the following definition: Two functions f and g are inverse functions if and only if both of their compositions are the identify function. That is,

[ f og](x)= x and [g o f ](x)= x.



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For the relation above, what restrictions could we place on the x’s or y’s so that the relation would be a function of x? Restriction on x-axis Restriction on y-axis Sketch of Function

x > -2 y >1.5



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Teaching Transparency 5 Solutions

For the relation above, what restrictions could we place on the x’s or y’s so that the relation would be a well-defined function of x?

Restriction on x-axis Restriction on y-axis Sketch of Function x > -2 y > 1.5

-2 < x < 1 .5 < y < 1.5



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-2 < x < 1 -.5 < y < .5

x > -2 y < -.5



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TASK 1.4.1: INVERSES, RELATIONS, AND FUNCTIONS Part I: Introduction On the following chart, give an example of a function for each representation listed. Function 1 Verbal Representation:

Function 2 Graphical Representation:

Function 3 Tabular Representation:

Function 4 Symbolic Representation:



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Part II. 1. For the graphs of each of the relations given, list what kinds of

restrictions we would have to place on the x’s or y’s or both so that the relations will be functions of x. Fill in the given table with at least two ways (if possible) to restrict the x’s or y’s or both so that the result will give a function of x.

Relation Restriction

on x-axis Restriction on y-axis

Sketch of function of x

a)

b)



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Relation Restriction on x-axis

Restriction on y-axis

Sketch of function of x

c)

d)



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Relation Restriction

on x-axis Restriction on y-axis

Sketch of function of

x e)

f)



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2. The graphs of each of the relations from Exercise 1 are given below. For each, sketch the inverse relation using patty paper as a guide. Which (if any) of these inverse relations are functions?

Relation Inverse Relation

a

b



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c

d



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e

f



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3. In the table below, the graphs of relations a, b, c, e, and f from Exercise 1 are given along with their respective inverse relations. The original relation is given as a solid curve and its inverse relation as a dotted curve. Also, portions of the original relation are shaded to indicate those portions that, under appropriate restrictions of x’s and/or y’s, are functions.

Fill in the given table. Explain the relationship between the domain and range of a given function and the domain and range of its inverse function. Include an example in your explanation.

Graphs


Restriction of relation on x-axis

Restriction of relation on y-axis







a)



Indicated by lighter (dotted) outline:




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Graphs






b)









c)







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Indicated by lightest (dotted) outline:










e)







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f)





4. Explain the “vertical line test” and the “horizontal line test” that are

used when referring to functions. Use the graphs in Exercise 3 in your explanation.

Documents

TASK 1.4.1: INVERSES RELATIONS AND FUNCTIONS 3 3 2 Function 4 Symbolic Representation: ... In Exercise 3, ... that the vertical line test helps determine if a relation is a function