Algebra II: Strand 1. Foundations of Functions; Topic 4. Making Connections; Task 1.4.1
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:
Function 2
Graphical Representation:
The candy bar that I get from the
vending machine depends upon
the code that I enter when
prompted.
Function 3
Tabular Representation:
x
-2
-1
0
1
2
3
y
1
2
1
2
3
2
Function 4
Symbolic Representation:
y=x2
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.
1
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Algebra II: Strand 1. Foundations of Functions; Topic 4. Making Connections; Task 1.4.1
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
a)
Restriction of
relation on
x-axis
Indicated by
darker outline:
Restriction of
relation on
y-axis
Indicated by
darker outline:
x>1
y>0
Domain of
inverse function
Range of inverse
function
Indicated by
darker (dotted)
outline:
Indicated by
darker (dotted)
outline:
x>0
y>1
December 10, 2004. Ensuring Teacher Quality: Algebra II, produced by the Charles A. Dana Center at The University of Texas
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Algebra II: Strand 1. Foundations of Functions; Topic 4. Making Connections; Task 1.4.1
Graphs
Relation—solid graph
Inverse Relation—dotted graph
b)
Restriction of
relation on
x-axis
Indicated by
lighter outline:
Restriction of
relation on
y-axis
Indicated by
lighter outline:
Domain of
inverse function
Range of inverse
function
Indicated by
lighter (dotted)
outline:
Indicated by
lighter (dotted)
outline:
x>1
y<0
x<0
y>1
Indicated by
darker outline:
Indicated by
darker outline:
Indicated by
darker (dotted)
outline:
Indicated by
darker (dotted)
outline:
None
None
None
None
Indicated by
darker outline:
Indicated by
darker outline:
Indicated by
darker (dotted)
outline:
Indicated by
darker (dotted)
outline:
None
!1.6 < y < 1.6 !1.6 < x < 1.6
!" < x < "
c)
!" < x < "
!" < y < "
!" < x < "
!" < y < "
None
!" < y < "
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Algebra II: Strand 1. Foundations of Functions; Topic 4. Making Connections; Task 1.4.1
Graphs
Relation—solid graph
Inverse Relation—dotted graph
Restriction of
relation on
x-axis
Indicated by
lighter outline:
Restriction of
relation on
y-axis
Indicated by
lighter outline:
Domain of
inverse function
Range of inverse
function
Indicated by
lighter (dotted)
outline:
Indicated by
lighter (dotted)
outline:
None
!4.7 < y < !1.6
!4.7 < x < !1.6
None
Indicated by
lightest outline:
Indicated by
lightest outline:
Indicated by
lightest(dotted)
outline:
Indicated by
lightest(dotted)
outline:
None
1.6 < y < 4.7
1.6 < x < 4.7
None
Indicated by
darker outline:
Indicated by
darker outline:
Indicated by
darker (dotted)
outline:
Indicated by
darker (dotted)
outline:
x>-2
y>1.5
x>1.5
y>-2
Indicated by
lighter outline:
Indicated by
lighter outline:
Indicated by
lighter (dotted)
outline:
Indicated by
lighter (dotted)
outline:
x<1/2
y<-1/2
x<-1/2
y<1/2
Indicated by
lightest outline:
Indicated by
lightest outline:
Indicated by
lightest (dotted)
outline:
Indicated by
lightest (dotted)
outline:
-2<x<1/2
-1/2<y<3/2
-1/2<x<3/2
-2<y<1/2
!" < x < "
!" < x < "
e)
!" < y < "
!" < y < "
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Algebra II: Strand 1. Foundations of Functions; Topic 4. Making Connections; Task 1.4.1
Graphs
Relation—solid graph
Inverse Relation—dotted graph
f)
Restriction of
relation on
x-axis
Indicated by
darker outline:
Restriction of
relation on
y-axis
Indicated by
darker outline:
Domain of
inverse function
Range of inverse
function
Indicated by
darker (dotted)
outline:
Indicated by
darker (dotted)
outline:
x>1
y>0
x>0
y>1
Indicated by
lighter outline:
Indicated by
lighter outline:
Indicated by
lighter (dotted)
outline:
Indicated by
lighter (dotted)
outline
x>1
y<0
x<0
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 xvalue. 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.
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.
Algebra II: Strand 1. Foundations of Functions; Topic 4. Making Connections; Task 1.4.1
6
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.
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.
Algebra II: Strand 1. Foundations of Functions; Topic 4. Making Connections; Task 1.4.1
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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)}
December 10, 2004. Ensuring Teacher Quality: Algebra II, produced by the Charles A. Dana Center at The University of Texas
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Algebra II: Strand 1. Foundations of Functions; Topic 4. Making Connections; Task 1.4.1
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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
x
-4
-3
-3
-2
1
1
2
y
5
-4
-3
0
1
2
-4
Inverse of
Relation 1
x
5
-4
-3
0
1
2
-4
y
-4
-3
-3
-2
1
1
2
December 10, 2004. Ensuring Teacher Quality: Algebra II, produced by the Charles A. Dana Center at The University of Texas
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Instructor Transparency 3
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.
Algebra II: Strand 1. Foundations of Functions; Topic 4. Making Connections; Task 1.4.1
11
Graphically we see:
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at Austin for the Texas Higher Education Coordinating Board.
Algebra II: Strand 1. Foundations of Functions; Topic 4. Making Connections; Task 1.4.1
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at Austin for the Texas Higher Education Coordinating Board.
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Algebra II: Strand 1. Foundations of Functions; Topic 4. Making Connections; Task 1.4.1
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 o g](x) = x and [g o f ](x) = x.
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.
Algebra II: Strand 1. Foundations of Functions; Topic 4. Making Connections; Task 1.4.1
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Instructor Transparency 5
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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at Austin for the Texas Higher Education Coordinating Board.
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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
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.
Algebra II: Strand 1. Foundations of Functions; Topic 4. Making Connections; Task 1.4.1
-2 < x < 1
-.5 < y < .5
x > -2
y < -.5
16
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at Austin for the Texas Higher Education Coordinating Board.
Algebra II: Strand 1. Foundations of Functions; Topic 4. Making Connections; Task 1.4.1
TASK 1.4.1: INVERSES, RELATIONS,
17
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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Algebra II: Strand 1. Foundations of Functions; Topic 4. Making Connections; Task 1.4.1
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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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Algebra II: Strand 1. Foundations of Functions; Topic 4. Making Connections; Task 1.4.1
Relation
Restriction
on x-axis
Restriction
on y-axis
19
Sketch of
function of x
c)
d)
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at Austin for the Texas Higher Education Coordinating Board.
Algebra II: Strand 1. Foundations of Functions; Topic 4. Making Connections; Task 1.4.1
Relation
Restriction
on x-axis
Restriction
on y-axis
20
Sketch of
function of
x
e)
f)
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at Austin for the Texas Higher Education Coordinating Board.
Algebra II: Strand 1. Foundations of Functions; Topic 4. Making Connections; Task 1.4.1
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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
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.
Algebra II: Strand 1. Foundations of Functions; Topic 4. Making Connections; Task 1.4.1
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c
d
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Algebra II: Strand 1. Foundations of Functions; Topic 4. Making Connections; Task 1.4.1
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e
f
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Algebra II: Strand 1. Foundations of Functions; Topic 4. Making Connections; Task 1.4.1
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
Relation—solid graph
Inverse Relation—dotted graph
a)
Restriction
of relation
on x-axis
Restriction
of relation
on y-axis
Domain of
inverse
function
Range of
inverse
function
Indicated
by darker
outline:
Indicated
by darker
outline:
Indicated
by darker
(dotted)
outline:
Indicated
by darker
(dotted)
outline:
Indicated
by lighter
outline:
Indicated
by lighter
outline:
Indicated
by lighter
(dotted)
outline:
Indicated
by lighter
(dotted)
outline:
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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Algebra II: Strand 1. Foundations of Functions; Topic 4. Making Connections; Task 1.4.1
Restriction
of relation
on x-axis
Restriction
of relation
on y-axis
Domain of
inverse
function
b)
Indicated
by darker
outline:
Indicated
by darker
outline:
Indicated
by darker
(dotted)
outline:
Indicated
by darker
(dotted)
outline:
c)
Indicated
by darker
outline:
Indicated
by darker
outline:
Indicated
by darker
(dotted)
outline:
Indicated
by darker
(dotted)
outline:
Indicated
by lighter
outline:
Indicated
by lighter
outline:
Indicated
by lighter
(dotted)
outline:
Indicated
by lighter
(dotted)
outline:
Graphs
Relation—solid graph
Inverse Relation—dotted graph
Range of
inverse
function
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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Algebra II: Strand 1. Foundations of Functions; Topic 4. Making Connections; Task 1.4.1
Graphs
Relation—solid graph
Inverse Relation—dotted graph
e)
Restriction
of relation
on x-axis
Restriction
of relation
on y-axis
Domain of
inverse
function
Range of
inverse
function
Indicated
by lightest
outline:
Indicated
by lightest
outline:
Indicated
by lightest
(dotted)
outline:
Indicated
by lightest
(dotted)
outline:
Indicated
by darker
outline:
Indicated
by darker
outline:
Indicated
by darker
(dotted)
outline:
Indicated
by darker
(dotted)
outline:
Indicated
by lighter
outline:
Indicated
by lighter
outline:
Indicated
by lighter
(dotted)
outline:
Indicated
by lighter
(dotted)
outline:
Indicated
by lightest
outline:
Indicated
by lightest
outline:
Indicated
by lightest
(dotted)
outline:
Indicated
by lightest
(dotted)
outline:
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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Algebra II: Strand 1. Foundations of Functions; Topic 4. Making Connections; Task 1.4.1
Graphs
Relation—solid graph
Inverse Relation—dotted graph
f)
Restriction
of relation
on x-axis
Restriction
of relation
on y-axis
Domain of
inverse
function
Range of
inverse
function
Indicated
by darker
outline:
Indicated
by darker
outline:
Indicated
by darker
(dotted)
outline:
Indicated
by darker
(dotted)
outline:
Indicated
by lighter
outline:
Indicated
by lighter
outline:
Indicated
by lighter
(dotted)
outline:
Indicated
by lighter
(dotted)
outline:
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.
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.