Describing Relations Unit
(Level IV Academic Math)
NSSAL
(Draft)
C. David Pilmer
2009
(Last Updated: Dec, 2011)
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Department of Labour and Advanced Education.
The following are permitted to use and reproduce this resource for classroom purposes.
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Individuals, not including teachers or instructors, are permitted to use this resource for their own
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are they permitted to use this resource under the direction of a teacher or instructor at a learning
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Acknowledgments
The Adult Education Division would like to thank the following university professors for
reviewing this resource to ensure all mathematical concepts were presented correctly and in a
manner that supported our learners.
Dr. Robert Dawson (Saint Mary’s University)
Dr. Genevieve Boulet (Mount Saint Vincent University)
The Adult Education Division would also like to thank the following NSCC instructors for
piloting this resource and offering suggestions during its development.
Charles Bailey (IT Campus)
Elliott Churchill (Waterfront Campus)
Barbara Gillis (Burridge Campus)
Barbara Leck (Pictou Campus)
Suzette Lowe (Lunenburg Campus)
Floyd Porter (Strait Area Campus)
Brian Rhodenizer (Kingstec Campus)
Joan Ross (Annapolis Valley Campus)
Jeff Vroom (Truro Campus)
Table of Contents
Introduction …………………………………………………………………………………..
Negotiated Completion Date …………………………………………………………………
The Big Picture ………………………………………………………………………………
Course Timelines …………………………………………………………………………….
i
i
ii
iii
Relations and Functions ……………………………………………………………………...
Domain and Range …………………………………………………………………………...
Symmetry and Intercepts …………………………………………………………………….
Putting It Together …………………………………………………………………………...
1
7
18
22
Post-Unit Reflections ………………………………………………………………………... 32
Terminology …………………………………………………………………………………. 33
Answers ……………………………………………………………………………………… 34
Introduction
When you take chemistry there are some fundamental terms like valence electrons, atomic
number, and atomic mass that you are expected to learn and understand. Without knowing these
terms, it is impossible to communicate with other students and your instructor. Without knowing
these terms, it is difficult to learn new concepts. Similarly in math, there are some fundamental
terms we use to describe relationships between two sets of numbers that you will need to learn.
In this short unit you will learn about relations, functions, domain, range, intercepts, odd
functions and even functions. These concepts will be revisited regularly as you proceed through
subsequent units in this math course.
Negotiated Completion Date
After working for a few days on this unit, sit down with your instructor and negotiate a
completion date for this unit.
Start Date:
_________________
Completion Date:
_________________
Instructor Signature: __________________________
Student Signature:
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The Big Picture
The following flow chart shows the optional bridging unit and the eight required units in Level
IV Academic Math. These have been presented in a suggested order.
Bridging Unit (Recommended)
• Solving Equations and Linear Functions
Describing Relations Unit
• Relations, Functions, Domain, Range, Intercepts, Symmetry
Systems of Equations Unit
• 2 by 2 Systems, Plane in 3-Space, 3 by 3 Systems
Trigonometry Unit
• Pythagorean Theorem, Trigonometric Ratios, Law of Sines,
Law of Cosines
Sinusoidal Functions Unit
• Periodic Functions, Sinusoidal Functions, Graphing Using
Transformations, Determining the Equation, Applications
Quadratic Functions Unit
• Graphing using Transformations, Determining the Equation,
Factoring, Solving Quadratic Equations, Vertex Formula,
Applications
Rational Expressions and Radicals Unit
• Operations with and Simplification of Radicals and Rational
Expressions
Exponential Functions and Logarithms Unit
• Graphing using Transformations, Determining the Equation,
Solving Exponential Equations, Laws of Logarithms, Solving
Logarithmic Equations, Applications
Inferential Statistics Unit
• Population, Sample, Standard Deviation, Normal Distribution,
Central Limit Theorem, Confidence Intervals
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Course Timelines
Academic Level IV Math is a two credit course within the Adult Learning Program. As a two
credit course, learners are expected to complete 200 hours of course material. Since most ALP
math classes meet for 6 hours each week, the course should be completed within 35 weeks. The
curriculum developers have worked diligently to ensure that the course can be completed within
this time span. Below you will find a chart containing the unit names and suggested completion
times. The hours listed are classroom hours. In an academic course, there is an expectation that
some work will be completed outside of regular class time.
Unit Name
Minimum
Completion Time
in Hours
0
6
18
18
20
36
12
20
20
Total: 150 hours
Bridging Unit (optional)
Describing Relations Unit
Systems of Equations Unit
Trigonometry Unit
Sinusoidal Functions Unit
Quadratic Functions Unit
Rational Expressions and Radicals Unit
Exponential Functions and Logarithms Unit
Inferential Statistics Unit
Maximum
Completion Time
in Hours
20
8
22
20
24
42
16
24
24
Total: 200 hours
As one can see, this course covers numerous topics and for this reason may seem daunting. You
can complete this course in a timely manner if you manage your time wisely, remain focused,
and seek assistance from your instructor when needed.
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Relations and Functions
Every time you have drawn a graph, you have been visually representing the "relationship"
between two sets of numbers (typically x-values and y-values). These relationships are referred
to as binary relations or simply relations. Graphs are only one way to express a relation. We
can also use tables of values, ordered pairs, mapping diagrams, equations, and inequalities to
illustrate the relationship between two sets of numbers.
Table of Values
x
0
1
2
3
Ordered Pairs
Mapping Diagram
(0, 9), (1, 7),
(2, 5), (3, 3)
y
9
7
5
3
0
9
1
7
2
5
3
3
Equation &
Inequalities
y = −2 x + 9
y ≥ −3 x + 9
A function is a special type of relation. A relation is a function if for every element in the first
set (typically x), there is one, and only one, element of the second set (typically y).
Example 1:
Consider the following three mapping diagrams. Determine which of these relations are
functions.
Mapping Diagram A
Mapping Diagram B
Mapping Diagram C
0
9
0
9
0
9
1
7
1
7
1
7
2
5
2
5
2
5
3
3
3
3
Answers:
Mapping Diagram A
• The element 0 in the first set has one corresponding element in the second set, 9.
• The element 1 in the first set has one corresponding element in the second set, 7.
• The element 2 in the first set has one corresponding element in the second set, 5.
• The element 3 in the first set has one corresponding element in the second set, 3.
• Since each element in the first set has one corresponding element in the second set, we
can conclude that this is a function.
Mapping Diagram B
• The element 2 in the first set has two corresponding elements in the second set, 5 and 3.
• We can conclude that this is not a function.
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Mapping Diagram C
• The element 0 in the first set has one corresponding element in the second set, 9.
• The element 1 in the first set has one corresponding element in the second set, 7.
• The element 2 in the first set has one corresponding element in the second set, 5.
• The element 3 in the first set has one corresponding element in the second set, 5.
• It does not matter that two elements (2 and 3) in the first set map onto the same element
(5) in the second set. We can still conclude that this is a function.
Example 2:
Explain why each of the following relations is or is not a function.
(a) {(1, 1), (2, 2), (3, 3), (4, 4 )}
(b) {(1, 2), (2, 2), (3, 2 ), (4, 2 )}
(c) {(1, 1), (1, 2), (1, 3), (1, 4 )}
Answers:
(a) The element 1 in the first set has one corresponding element in the second set, 1.
The element 2 in the first set has one corresponding element in the second set, 2.
The element 3 in the first set has one corresponding element in the second set, 3.
The element 4 in the first set has one corresponding element in the second set, 4.
Since each member of the first set has only one corresponding member in the second set,
we can conclude that this relation is a function.
(b) The elements 1, 2, 3, and 4 in the first set have only one corresponding element in the
second set, 2. Since each element the first set has only one corresponding element in the
second set, we can conclude that this relation is a function.
(c) The element 1 in the first set has four corresponding elements in the second set, 1, 2, 3,
and 4. Since the only element in the first set has more than one corresponding element in
the second set, we can conclude that this relation is not a function.
Example 3:
Explain why each of the following relations is or is not a function.
(a)
(b)
6
4
5
3
4
2
3
2
1
1
0
-3
-2
-1
0
1
2
0
3
0
-1
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(c)
(d)
4
5
3
4
2
3
1
2
0
-1
-1 0
1
2
3
4
5
1
6
0
-2
-3
-3
-2
-1
-4
-1
0
1
2
3
-2
Answers:
(a) Since each element in the first set has only one corresponding element in the second set,
we can conclude that this graph is a function.
(b) Since the element 1 in the first set has three corresponding elements (3, 4, and 5) in the
second set and the element 2 in the first set has two corresponding elements (3 and 4) in
the second set, we can conclude that this graph is not a function.
(c) Since every element in the first set, with the exception of 0, has two corresponding
elements in the second set, we can conclude that this graph is not a function.
(d) Since each element in the first set has only one corresponding element in the second set,
we can conclude that this graph is a function.
Questions
1. Explain why each of the following relations is or is not a function.
(a)
Explanation:
9
5
7
6
5
3
(b)
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Explanation:
7
-1
10
-2
13
-3
16
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(c)
Explanation:
5
8
11
3
5
14
(d)
Explanation:
10
9
20
30
2. Explain why each of the following relations is or is not a function.
(a) {(4, 8), (5, 4 ), (6, 2 ), (7, 1), (8, 0.5)}
(b) {(4, 4), (5, 1), (6, 0), (7, 1), (8, 4 )}
Explanation:
Explanation:
(c) {(6, 4), (0, 1), (− 2, 0), (0, − 1), (6, -4 )}
Explanation:
(d) {(− 2, 3), (− 1, 5), (0, 7 ), (0, 9 ), (1, 11)}
Explanation:
3. Determine whether each of the following is a function.
(a)
x
y
(b)
x
4
10
-1
4
11
0
4
12
1
4
13
2
4
14
3
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y
2
0.5
0
0.5
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4. Explain why the equations y = 3 x + 5 and y = 2 are both functions.
5. Explain why each of the following relations is or is not a function.
(a)
(b)
9
8
7
6
5
6
5
4
3
4
3
2
1
0
2
1
0
0
1
2
3
4
5
6
0
(c)
1
2
3
(d)
2
4
3
1
2
1
0
-2
-1
0
1
2
3
4
5
-15
-1
-10
-5
0
-1 0
5
10
15
-2
-3
-2
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(e)
(f)
2
1
1
0
-3
-2
-1
0
0
1
2
3
-1
0
180
360
540
720
900
-1
-2
-2
-3
(g)
(h)
3
4
2
3
1
2
0
-5
-4
-3
-2
-1
0
1
-1
1
-2
0
0
-3
1
2
3
4
5
6. Determine whether the following verbal statements represent functions.
(a) y is greater than or equal to x.
_______________
(b) y is equal to x plus one.
_______________
(c) y is not equal to x.
_______________
For Your Information (FYI)
The Vertical Line Test: If a vertical line placed anywhere on the coordinate system intersects
the graph at more than one point, then the graph does not depict a function.
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Domain and Range
As we previously stated, a relation shows the relationship between two sets of numbers.
Sometimes in mathematics it is important to describe each of those sets of numbers separately.
The domain describes all the members in the first set (typically the x-values). The range
describes all the members in the second set (typically the y-values).
Example 1:
Determine the range and domain of the following relations.
(a)
(b)
9
8
7
6
5
2
1
4
3
2
1
0
0
-2
-1
0
1
2
3
4
5
-1
0
1
2
3
-2
Answers:
With both of these graphs we are dealing with discrete points. These points are not
connected together by a line. When stating the domain and range of graphs comprised of
discrete points, it is just a matter of listing the numbers in the first set and the numbers in the
second set.
(a) We have four order pairs: (0, 1), (1, 2), (2, 4), and (3, 8).
Our four first set values are 0, 1, 2, and 3.
Therefore: Domain: {0, 1, 2, 3}
Our four second set values are 1, 2, 4, and 8.
Therefore: Range: {1, 2, 4, 8}
(b) We have eight order pairs: (-2, -1), (-1, 0), (0, 1), (1, 0), (2, -1), (3, 0), (4, 1) and (5, 0).
Our four first set values are -2, -1, 0, 1, 2, 3, 4, 5.
Therefore: Domain: {-2, -1, 0, 1, 2, 3, 4, 5}
Our second set values are -1, 0, 1.
Therefore: Range: {-1, 0, 1}
At this point we should take a few minutes to look at the real number system. Real numbers are
any number that can be plotted on a number line. Up to this point in our mathematical
educations, that is every type of number that we have encountered. Therefore real numbers
include fractions, decimals, whole numbers, positive numbers, negative numbers, repeating
decimals, roots (e.g. 2 , 3 11 ), and pi ( π = 3.14159... ). The symbol R is used to represent real
numbers. It may be hard to believe but there are numbers that are not real but we will not be
encountering them in this course. If we took higher level math courses or pursued careers in
electronics, we would have to work with these non-real numbers.
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All real numbers can be divided into one of two categories; rational numbers or irrational
numbers. Rational numbers are numbers that can be expressed as fractions using two integers.
Some examples of rational numbers have been supplied below.
23 =
23
1
The number 23 is the fraction comprised of the integers 23 and 1.
−6=
−6
1
The number -6 is the fraction comprised of the integers -6 and 1
(or 6 and -1).
0.666... =
2
3
The repeating decimal 0.666… is the fraction comprised of the integers 2
and 3.
− 0.07 =
−7
100
The number -0.07 is the fraction comprised of the integers of -7 and 100
(or 7 and -100).
5
12
9 =3=
The number
3
1
5
is already a fraction.
12
The number 9 can be expressed as the fraction comprised of the integers
3 and 1. Please note that most roots can not be simplified in this manner
and are therefore classified as irrational numbers.
Irrational numbers are numbers that cannot be expressed as the ratio of two integers. This
means that we are dealing the constant pi ( π = 3.14159... ) and roots (e.g. 7 , 5 2 , 3 10 ) that
cannot be expressed as a rational number.
The above information can be summarized in the following diagram. Please take note of the
symbols used to represent the different types of numbers.
Rational
Numbers (Q)
Real
Numbers (R)
Irrational
Numbers ( Q )
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Natural Numbers (N): 1, 2, 3, 4,…
Whole Numbers (W): 0, 1, 2, 3, 4,…
Integers (I): …,-2, -1, 0, 1, 2,…
Common Fractions
Repeating and Terminating Decimals
e.g.
8
2 , 4 3 , 3 12 , π
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So why did we have to learn about the real number system in the middle of a section concerned
with range and domain? Consider the answer for the domain in Example 1(b). The domain was
stated in the following manner.
Domain: {-2, -1, 0, 1, 2, 3, 4, 5}
There is another way this domain can be stated using our knowledge of the real number system.
All these values of x are consecutive integers where the smallest value is -2 and the largest value
is 5. Therefore the domain can also be written in the following manner.
Domain: {xεI - 2 ≤ x ≤ 5 }
This is read as “x is a member of integers such that x is greater than or equal to -2 and less than
or equal to 5.” The epsilon ( ε ) means “a member of.” The vertical line means “such that.”
Example 2:
Determine the range and domain of the following mapping diagram.
Answer:
This type of question is handled in the same manner as graphs
comprised of discrete points.
Domain: {0, 1, 2, 3} or {xεI 0 ≤ x ≤ 3 } or {xεW 0 ≤ x ≤ 3 }
Range: {1, 3, 5, 7, 9}
0
9
1
7
2
5
3
3
1
Example 3:
Determine the range and domain of the following relations. Assume that the x-axis and y-axis
have been labeled.
Answers:
(a)
The solid dot indicates an endpoint. The arrow
indicates that the graph continues beyond the
scale that was used. The x-values go from 1 to
positive infinity. Since we are dealing with a
solid line rather than discrete points, then we
are not restricted to whole numbers. We know
that we are dealing with real numbers.
Therefore the domain is any real number
greater than or equal to 1.
Domain: {xεR x ≥ 1 }
The y-values go from 2 to positive infinity.
Therefore the range is any real number greater
than or equal to 2.
Range: {yεR y ≥ 2 }
6
5
4
3
2
1
0
0
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Answers:
(b)
5
4
3
2
1
0
-1
-1 0
1
2
3
4
5
6
-2
-3
This particular graph has two endpoints. It
terminates at (2, 4) and (5, -2). The x-values go
from 2 to 5. Therefore the domain is any real
number that is greater than or equal to 2 and
less than or equal to 5.
Domain: {xεR 2 ≤ x ≤ 5 }
The y-values go from -2 to 4. Therefore the
range is any real number that is greater than or
equal to -2 and less than or equal to 4.
Range: {yεR - 2 ≤ y ≤ 4 }
(c)
4
3
2
1
0
-1
-1 0
1
2
3
4
5
6
-2
The x-values go from 0 to positive infinity.
Therefore the domain is any real number
greater than or equal to 0
Domain: {xεR x ≥ 0 }
The y-values go from negative infinity to
positive infinity. Therefore the range is any
real number.
Range: {yεR }
-3
-4
(d)
1
0
-3
-2
-1
0
-1
-2
1
2
3
The x-values go from negative infinity to
positive infinity. Therefore the domain is any
real number.
Domain: {xεR }
It does not matter what x-value one chooses,
the y-value is always equal to -2. Since there
is only one value in the second set, you just
state that value.
Range: {− 2}
-3
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Answers:
(e)
5
4
3
2
1
0
-1 0
-2
1
2
3
4
5
The x-values go from negative infinity to 5.
Therefore the domain is any real number less
than or equal to 5.
Domain: {xεR x ≤ 5 }
The y-values go from negative infinity to 4.
Therefore the range is any real number that is
less than or equal to 4.
Range: {yεR y ≤ 4 }
-3
-4
-5
(f)
The x-value is always equal to -3. Since there
is only one value in the first set, you just state
that value.
Domain: {− 3}
The y-values go from 2 to 5. Therefore the
range is any real number that is greater than or
equal to 2 and less than or equal to 5.
Range: {yεR 2 ≤ y ≤ 5 }
6
5
4
3
2
1
0
-4
-3
-2
-1
-1
0
1
Questions:
1. A relation is defined by the following ordered pairs. State the range and domain.
{(1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6)}
2. A relation is defined by the following mapping rule. State the
range and domain.
3
5
1
7
0
9
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3. Determine the range and domain of the following relations. Assume that the x-axis and yaxis have been labeled.
(a)
(b)
6
4
5
3
4
2
3
2
1
1
0
-3
-2
-1
0
1
2
0
3
0
-1
1
Domain:
Domain:
Range:
Range:
(c)
(d)
9
8
7
6
5
3
4
3
2
1
0
4
3
2
1
0
-1
-1
0
1
2
3
4
5
-2
-3
0
1
2
3
4
-4
Domain:
Domain:
Range:
Range:
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(e)
(f)
5
4
3
1
0
-3
-2
-1
0
1
2
3
2
1
0
-1 -1 0
-2
-1
-2
-2
1
2
3
4
5
6
-3
-4
-5
-3
-4
Domain:
Domain:
Range:
Range:
(g)
(h)
6
4
5
3
2
4
1
3
2
-5
-4
-3
-2
0
-1 -1 0
1
1
2
3
4
5
-2
-3
0
0
1
2
3
4
5
6
-4
Domain:
Domain:
Range:
Range:
4. Create your own relation comprised of discrete
points. Now state the domain and range of your
relation.
4
3
2
1
0
-4
-3
-2
-1
0
1
2
3
4
-1
-2
-3
-4
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5. Determine the range and domain of the following relations. Assume that the x-axis and yaxis have been labeled.
(a)
(b)
-3
-2
6
6
5
5
4
4
3
3
2
2
1
1
0
-1 -1 0
1
2
3
4
5
-3
-2
-2
Domain:
Range:
Range:
(c)
(d)
-2
6
6
5
5
4
4
3
3
2
2
1
1
0
-1 -1 0
1
2
3
4
5
1
2
3
4
5
-2
Domain:
-3
0
-1 -1 0
1
2
3
4
5
-3
-2
-2
0
-1 -1 0
-2
Domain:
Domain:
Range:
Range:
6. Determine the domain and range of y = 2 x − 1 . Explain how you arrived at your answer.
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7. Determine the range and domain of the following relations. Assume that the x-axis and yaxis have been labeled.
(a)
(b)
3
2
1
-1
0
-5
-4
-3
-2
-1
0
1
2
3
4
5
-1
-2
-3
4
3
2
1
0
-1 0
-2
-3
-4
-5
-6
-7
Domain:
Domain:
Range:
Range:
(c)
(d)
1
2
3
4
6
4
5
3
4
2
3
1
2
0
-6
-5
-4
-3
-2
-1
1
-2
-1
6
0
1
-2
0
-3
-1
5
0
1
2
3
4
5
-3
Domain:
Domain:
Range:
Range:
4
8. Draw a linear function with the following domain
and range. There are two acceptable answers for
this question.
Domain: {xεR - 1 ≤ x ≤ 3 }
3
2
1
Range: {yεR - 4 ≤ y ≤ 2 }
0
-4
-3
-2
-1
0
1
2
3
4
-1
-2
-3
-4
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9. Determine the range and domain of the following relations. Assume that the x-axis and yaxis have been labeled.
(a)
(b)
9
5
8
4
7
3
6
2
5
4
1
3
0
-3
-2
-1
-1
2
0
1
2
3
1
0
-2
-4
-3
-2
Domain:
Domain:
Range:
Range:
(c)
(d)
-1
0
1
2
3
4
5
6
3
7
6
2
5
1
4
0
3
-5
2
-4
-3
-2
-1
0
1
-1
1
-2
0
-6
-5
-4
-3
-2
-1 -1 0
1
2
-3
Domain:
Domain:
Range:
Range:
10. Draw a nonlinear function with the following domain
and range. There is more than one acceptable
answers for this question.
Domain: {xεR x ≥ −2 }
4
3
2
Range: {yεR - 1 ≤ y ≤ 3 }
1
0
-4
-3
-2
-1
0
1
2
3
4
-1
-2
-3
-4
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11. Determine the range and domain of the following relations. Assume that the x-axis and yaxis have been labeled.
(a)
(b)
8
9
8
7
6
5
7
6
5
4
4
3
2
1
0
3
2
1
0
0
10
20
30
40
50
60
70
-1
0
Domain:
Domain:
Range:
Range:
(c)
(d)
6
35
5
30
1
2
3
4
5
25
4
20
3
15
2
10
1
5
0
0
0
1
2
3
4
0
5
Domain:
Domain:
Range:
Range:
1
2
3
4
12. The equation d = 2t + 1 describes the distance, d, an object travels in terms of time, t. We
are told that the equation is only applicable from t = 0 seconds to t = 3 seconds. State the
domain and range of the function.
13. The equation c = 15n + 30 describes the cost, c, of purchasing gravel with respect to the
number, n, of tons purchased. The equation only applies from n = 0 to n =10. State the
domain and range of the function.
NSSAL
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Intercepts and Symmetry
The x-intercept is the x-coordinate of the point where a graph intersects the x-axis. The yintercept is the y-coordinate of the point where a graph intersects the y-axis.
Example 1
Determine the x-intercept(s) and y-intercept(s) for each of the following relations. Assume that
the x-axis and y-axis have been labeled.
(a)
(b)
5
6
4
5
4
3
3
2
2
1
1
0
-3
-2
-1
-1
0
1
2
3
-3
0
-1 -1 0
-2
-2
1
2
3
4
5
-2
Answers:
x-intercept: -2 and 2
y-intercept: 4
Answers:
x-intercept: 3
y-intercept: 3
(c)
(d)
7
3
6
2
5
1
4
3
0
-5
-4
-3
-2
-1
0
1
2
-1
1
-2
0
-10
-3
Answers:
x-intercept: -1
y-intercept: none
NSSAL
©2009
-8
-6
-4
-2
0
2
4
6
8
10
Answers:
x-intercept: -4 and 4
y-intercept: 6
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Important Note:
The terms odd and even are used to describe the symmetry of graphs of relations. Although
these terms are used with all relations, in this course we are only going to apply them to
functions. Courses at this level typically only apply these terms to functions.
Geometrically, the graph of an even function is symmetrical about the y-axis. This means that
graph remains unchanged after a reflection in the y-axis. The four graphs shown below are all
examples of even functions.
Geometrically, the graph of an odd function has rotational symmetry about the origin. This
means that the graph remains unchanged after a rotation of 180o about the origin. The four
graphs shown below are all examples of odd functions.
Most functions are neither odd nor even. Such is the case with the functions below.
Sometimes placing restrictions on the
domain of a function can change its
symmetry. For example, the first function
shown on the right has the equation
y = − x 2 + 6 . It is an even function. If,
however, we restrict the domain such that
all x-values must be greater than or equal to
-2, then the resulting graph is neither odd
nor even.
NSSAL
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y = −x2 + 6
y = − x 2 + 6, x ≥ −2
Even
Neither
Draft
C. D. Pilmer
For Your Information (FYI)
In this course we determine if a function is odd or even by looking at the graph. If
you decide to pursue higher level math courses, they will expand upon our definition
to include an algebraic description of odd and even functions. A function, f, is said to
be even if f (a ) = f (− a ) for all values in the domain. A function, f, is said to be odd
if f (a ) = − f (− a ) for all values in the domain. You are not expected to know this for
this particular course.
Questions:
1. Determine the x-intercept(s) and y-intercept(s) for each of the following relations. If we are
dealing with a function, classify it as odd, even, or neither. The following graphs were drawn
using a graphing calculator where the scale goes from -5 to 5 on both the x-axis and y-axis.
(a)
(b)
(c)
x-intercept:
x-intercept:
x-intercept:
y-intercept:
y-intercept:
y-intercept:
symmetry:
symmetry:
symmetry:
(d)
(e)
(f)
x-intercept:
x-intercept:
x-intercept:
y-intercept:
y-intercept:
y-intercept:
symmetry:
symmetry:
symmetry:
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(g)
(h)
(i)
x-intercept:
x-intercept:
x-intercept:
y-intercept:
y-intercept:
y-intercept:
symmetry:
symmetry:
symmetry:
(j)
(k)
(l)
x-intercept:
x-intercept:
x-intercept:
y-intercept:
y-intercept:
y-intercept:
symmetry:
symmetry:
symmetry:
(m)
(n)
(o)
x-intercept:
x-intercept:
x-intercept:
y-intercept:
y-intercept:
y-intercept:
symmetry:
symmetry:
symmetry:
NSSAL
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Putting It Together
To recap:
• A function is a relation in which each element in the first set (x-values) has one, and only
one, corresponding element in the second set (y-values).
• The domain of a relation is all the members of the first set.
• The range of a relation is all the members of the second set.
• The x-intercept is the x-coordinate of the point where a graph intersects the x-axis.
• The y-intercept is the y-coordinate of the point where a graph intersects the y-axis.
• For this course we will only be using the terms odd and even when describing functions.
A function is even if it is symmetrical about the y-axis. A function is odd if it possesses
180o rotational symmetry about the origin.
Example 1:
For each of the graphs, answer the following questions.
(a) Are we dealing with a function? Explain.
(b) State the domain and range.
(c) State the x-intercept(s) and the y-intercept(s).
(d) If we are dealing with a function, state whether it is odd, even, or neither?
Graph I
3
2
1
0
-5
-4
-3
-2
-1
0
1
2
3
4
5
-1
-2
-3
Answers for Graph I:
(a) This is a function because it passes the vertical line test. That means that for every
element in the first set (x-values) there is only one corresponding element in the second
set (y-values).
(b) The x-values go from negative infinity to positive infinity therefore we would write the
domain as follows.
Domain: {xεR }
The y-values go from negative infinity to positive infinity therefore we write the range as
follows.
Range: {yεR }
(c) x-intercept: 0
y-intercept: 0
o
(d) Since the function has 180 rotational symmetry about the origin then the function is odd.
NSSAL
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Graph II
8
7
6
5
4
3
2
1
0
-2
-1
0
1
2
3
4
-1
-2
Answers for Graph II:
(a) This is a function because it passes the vertical line test. That means that for every
element in the first set (x-values) there is only one corresponding element in the second
set (y-values).
(b) The x-values go from negative infinity to positive infinity therefore we would write the
domain as follows.
Domain: {xεR }
The y-values go from -1 to positive infinity therefore we write the range as follows.
Range: {yεR y ≥ −1 }
(c) x-intercepts: 0 and 2
y-intercept: 0
(d) The function is neither odd nor even.
Graph III
4
3
2
1
-2
0
-1 -1 0
1
2
3
4
5
6
7
8
9
-2
-3
-4
Answers for Graph III:
(a) This is not a function because it fails the vertical line test. In this case, most elements in
the first set (x-values) have two corresponding elements in the second set (y-values).
(b) The x-values go from -1 to 8 therefore we would write the domain as follows.
Domain: {xεR - 1 ≤ x ≤ 8 }
The y-values go from -3 to 3 therefore we write the range as follows.
Range: {yεR - 3 ≤ y ≤ 3 }
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C. D. Pilmer
(c) x-intercept: -1
y-intercepts: -1 and 1
(d) We will not classify as odd, even, or neither since we are not dealing with a function.
Graph IV
7
6
5
4
3
2
1
0
0
5
10 15 20 25 30 35 40 45 50 55 60 65
Answers for Graph IV:
(a) This is a function because it passes the vertical line test. That means that for every
element in the first set (x-values) there is only one corresponding element in the second
set (y-values).
(b) The x-values go from zero to positive infinity therefore we would write the domain as
follows.
Domain: {xεR x ≥ 0 }
The y-values go from 4 to 6 therefore we write the range as follows.
Range: {yεR 4 ≤ y ≤ 6 }
(c) x-intercept (none)
y-intercept = 4
(d) If the function existed for values less than zero, then it would be classified as even. It,
however, has an endpoint at (0, 4). For this reason the function is neither odd nor even.
Graph V
3
2
1
0
-6 -5 -4 -3 -2 -1 0
-1
1
2
3
4
5
6
-2
-3
NSSAL
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C. D. Pilmer
Answers for Graph V:
(a) This is a function because it passes the vertical line test. That means that for every
element in the first set (x-values) there is only one corresponding element in the second
set (y-values).
(b) The x-values go from negative infinity to positive infinity therefore we would write the
domain as follows.
Domain: {xεR }
The y-values go from negative infinity to 2 therefore we write the range as follows.
Range: {yεR y ≤ 2 }
(c) x-intercepts: 3 and -3
y-intercept = 2
(d) Since this function is symmetrical about the y-axis (i.e. the y-axis cuts the function in
half), we classify it as even.
Graph VI
3
2
1
0
-1
0
1
2
3
4
-1
-2
Answers for Graph VI
(a) This is not a function because it fails the vertical line test. In this case, the only element
in the first set (x = 3) has many corresponding elements in the second set (y-values).
(b) The only x-value is 3 therefore we would write the domain as follows.
Domain: {3 }
The y-values go from -1 to 2 therefore we write the range as follows.
Range: {yεR - 1 ≤ y ≤ 2 }
(c) x-intercept: 3
y-intercept: none
(d) We will not classify as odd, even, or neither since we are not dealing with a function.
NSSAL
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Graph VII
4
3
2
1
0
0
1
2
3
4
5
6
7
Answers for Graph VII:
(a) This is a function because it passes the vertical line test. That means that for every
element in the first set (x-values) there is only one corresponding element in the second
set (y-values).
(b) Domain: {xεR 0 ≤ x ≤ 6 }
Range: {yεR 0 ≤ y ≤ 3 }
(c) x-intercept: 6
y-intercept = 1
(d) Neither odd nor even
Questions:
1. For each of the following graphs, determine if we are dealing with a function, state the
domain, range, and intercepts, and if we are dealing with a function state whether it is odd,
even or neither.
Graph I
Function or Not a Function (circle)
5
4
3
-3
-2
-1
2
1
0
-1 0
-2
-3
-4
-5
NSSAL
©2009
Domain
Range
1
2
3
x-intercepts
y-intercepts
Odd, Even, or Neither
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Graph II
Function or Not a Function (circle)
5
Domain
4
3
Range
2
1
x-intercepts
0
-6
-5
-4
-3
-2
-1 -1 0
1
2
3
y-intercepts
-2
Odd, Even, or Neither
-3
Graph III
Function or Not a Function (circle)
-2
-1
5
4
3
2
1
0
-1 0
-2
-3
-4
-5
-6
-7
Domain
Range
1
2
3
4
5
x-intercepts
y-intercepts
Odd, Even, or Neither
Graph IV
Function or Not a Function (circle)
5
4
3
-3
-2
-1
2
1
0
-1 0
-2
-3
-4
-5
NSSAL
©2009
Domain
Range
1
2
3
x-intercepts
y-intercepts
Odd, Even, or Neither
27
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Graph V
Function or Not a Function (circle)
8
7
Domain
6
Range
5
4
x-intercepts
3
2
y-intercepts
1
0
-5
-4
-3
-2
-1
0
1
2
3
4
5
Odd, Even, or Neither
Graph VI
Function or Not a Function (circle)
-4
-3
6
5
4
3
2
1
0
-1 -1 0
-2
-3
-4
-5
-6
-2
Domain
Range
1
2
3
4
x-intercepts
y-intercepts
Odd, Even, or Neither
Graph VII
Function or Not a Function (circle)
5
Domain
4
3
Range
2
x-intercepts
1
y-intercepts
0
-5
-4
NSSAL
©2009
-3
-2
-1
0
1
2
3
4
5
Odd, Even, or Neither
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C. D. Pilmer
Graph VIII
Function or Not a Function (circle)
0
-5
-4
-3
-2
-1
0
1
2
3
4
5
-1
Domain
-2
Range
-3
x-intercepts
-4
y-intercepts
-5
Odd, Even, or Neither
-6
Graph IX
Function or Not a Function (circle)
4
Domain
3
2
Range
1
0
-15
-10
-5
-1 0
5
10
15
x-intercepts
y-intercepts
-2
-3
Odd, Even, or Neither
-4
Graph X
Function or Not a Function (circle)
1
-4
-3
-2
0
-1 -1 0
-2
-3
Domain
1
2
3
4
Range
x-intercepts
-4
-5
y-intercepts
-6
-7
NSSAL
©2009
Odd, Even, or Neither
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C. D. Pilmer
Graph XI
Function or Not a Function (circle)
3
Domain
2
1
Range
0
-4
-3
-2
-1
-1
0
1
2
3
4
5
6
-2
x-intercepts
y-intercepts
-3
Odd, Even, or Neither
-4
Graph XII
Function or Not a Function (circle)
4
3
Domain
2
Range
1
-3
-2
0
-1 0
-1
1
x-intercepts
-2
y-intercepts
-3
-4
Odd, Even, or Neither
-5
Graph XIII
Function or Not a Function (circle)
5
Domain
4
3
Range
2
1
x-intercepts
0
-1 -1 0
1
2
3
4
5
6
7
8
y-intercepts
-2
-3
NSSAL
©2009
Odd, Even, or Neither
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4
3
Distance in Metres
2. Hiroshi’s son is operating a remote control truck in
front of a motion detector. His son is merely
driving towards or away from the detector. He is
not moving the truck from left to right, or visa
versa. The motion detector records the distance
between itself and the truck at regular time
intervals. The resulting graph shows the
relationship between this distance, d, and time, t.
(a) Does this graph depict a function?
(b) State the domain and range.
(c) Determine the t-intercept and d-intercept.
(d) Is the relation odd, even or neither?
2
1
0
0
1
2
3
4
5
6
Time in Seconds
140
Concentration in Bacteria per Square Centimetre
3. A bacteria population is growing in a Petri dish.
The concentration, c, of bacteria per square
centimeter is on the vertical axis. The time, t, in
hours is on the horizontal axis.
(a) Does this graph depict a function?
(b) State the domain and range.
(c) Determine the t-intercept and c-intercept.
(d) Is the relation odd, even or neither?
130
120
110
100
90
80
70
60
50
40
30
20
10
0
0
1
2
3
4
Time in Hours
NSSAL
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Post-Unit Reflections
What is the most valuable or important
thing you learned in this unit?
What part did you find most interesting or
enjoyable?
What was the most challenging part, and
how did you respond to this challenge?
How did you feel about this math topic
when you started this unit?
How do you feel about this math topic
now?
Of the skills you used in this unit, which
is your strongest skill?
What skill(s) do you feel you need to
improve, and how will you improve them?
How does what you learned in this unit fit
with your personal goals?
NSSAL
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Terminology
In this unit, you were exposed to the following terminology. These terms are presented in the
order they appear in this resource.
Relation
Function
Domain
Range
Real Numbers
Rational Numbers
Irrational Numbers
Natural Numbers
Whole Numbers
Integers
x-intercept
y-intercept
Even Function
Odd Function
NSSAL
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Answers
Relations and Functions (pages 1 to 6)
1. (a) Not a Function
(c) Function
(b) Function
(d) Not a Function
2. (a) Function
(c) Not a Function
(b) Function
(d) Not a Function
3. (a) Not a Function
(b) Function
4. They are both functions because when you put a single value in for x into the equation, only
one value of y is generated. For example with the equation y = 3 x + 5 if you substitute the xvalue of -3 into the equation, then the only one y-value (-4) is generated by the equation. The
equation y = 2 produces a graph that is a horizontal straight line. For this equation,
regardless of the x-value one chooses, the only corresponding y-value is two.
5. (a)
(c)
(e)
(g)
Not a Function
Function
Function
Not a Function
(b)
(d)
(f)
(h)
6. (a) Not a Function
(c) Not a Function
Function
Not a Function
Function
Not a Function
(b) Function
Domain and Range (pages 7 to 17)
1. Domain: {1}
Range: {1, 2, 3, 4, 5, 6} or {yεN 1 ≤ y ≤ 6 } or {yεW 1 ≤ y ≤ 6 } or {yεI 1 ≤ y ≤ 6 }
2. Domain: {3, 5, 7, 9}
Range: {1, 0}
3. (a) Domain: {-2, -1, 0, 1, 2} or {xεI - 2 ≤ x ≤ 2 }
Range: {3}
(b) Domain: {1, 2, 3} or {xεW 1 ≤ x ≤ 3 } or {xεN 1 ≤ x ≤ 3 } or {xεI 1 ≤ x ≤ 3 }
Range: {3, 4, 5} or {yεW 3 ≤ y ≤ 5 } or {yεN 3 ≤ y ≤ 5 } or {yεI 3 ≤ y ≤ 5 }
(c) Domain: {0, 1, 2, 3, 4} or {xεW 0 ≤ x ≤ 4 } or {xεI 0 ≤ x ≤ 4 }
Range: {0, 1, 4, 9}
(d) Domain: {4}
Range: {-3, -2, -1, 0, 1, 2} or {yεI - 3 ≤ y ≤ 2 }
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(e) Domain: {-2, -1, 0, 1, 2} or {xεI - 2 ≤ x ≤ 2 }
Range: {-1, -2, -3}
(f) Domain: {-2, 0, 2, 4, 6}
Range: {-4, -3, -2, -1, 0, 1, 2, 3, 4}
(g) Domain: {1, 2, 3, 4, 5} or {xεN 1 ≤ x ≤ 5 } or {xεW 1 ≤ x ≤ 5 } or {xεI 1 ≤ x ≤ 5 }
Range: {1, 2, 3, 4, 5} or {yεW 1 ≤ y ≤ 5 } or {yεN 1 ≤ y ≤ 5 } or {yεI 1 ≤ y ≤ 5 }
(h) Domain: {-3, 0, 3)
Range: {-2, -1, 0, 1, 2} or {yεI - 2 ≤ y ≤ 2 }
4. Answers will vary.
5 (a) Domain: {xεR }
Range: {yεR }
(b) Domain: {xεR x ≥ −2 }
(c) Domain: {xεR x ≤ 4 }
Range: {yεR y ≤ 5 }
Range: {yεR y ≥ −1 }
Range: {yεR - 1 ≤ y ≤ 5 }
(d) Domain: {xεR - 2 ≤ x ≤ 4 }
6. This is a linear function. Since no endpoints have been given we can assume that the graph
extends in both directions.
Domain: {xεR } Range: {yεR }
7. (a) Domain: {xεR x ≥ −4 }
(b) Domain: {xεR 1 ≤ x ≤ 5 }
(c) Domain: {xεR x ≤ 4 }
Range: {yεR y ≥ −2 }
Range: {yεR - 6 ≤ y ≤ 2 }
Range: {yεR y ≤ 5 }
(d) Domain: {xεR - 5 ≤ x ≤ −1 }
Range: {yεR - 1 ≤ y ≤ 3 }
8. Two options:
A line segment that goes from (-1, 2) to (3, -4)
or
A line segment that goes from (-1, -4) to (3, 2)
9. (a) Domain: {xεR }
Range: {yεR y ≤ 4 }
(b) Domain: {xεR }
Range: {6}
(c) Domain: {− 4}
Range: {yεR 1 ≤ y ≤ 6 }
(d) Domain: {xεR x ≤ −1 } Range: {yεR }
10. Answers will vary.
11. (a) Domain: {xεR x ≥ 0 }
NSSAL
©2009
Range: {yεR 1 ≤ y ≤ 7 }
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(b) Domain: {xεR x ≤ 3 }
(c) Domain: {xεR 1 ≤ x ≤ 4 }
Range: {yεR y ≥ 1 }
(d) Domain: {xεR 0 ≤ x ≤ 4 }
12. Domain: {tεR 0 ≤ t ≤ 3 }
13. Domain: {nεR 0 ≤ n ≤ 10 }
Range: {yεR 1 ≤ y ≤ 5 }
Range: {yεR 10 ≤ y ≤ 30 }
Range: {dεR 1 ≤ d ≤ 7 }
Range: {cεR 30 ≤ c ≤ 180 }
Intercepts and Symmetry (pages 18 to 21)
1.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
(j)
(k)
(l)
(m)
(n)
(o)
x-intercepts
-2 and 2
0
-3 and 1
-2
-3, -1, 1, and 3
0 and 4
-3, 0, and 3
None
-3, -1, 1, 3
None
-4, 0, and 4
3
-2 and 2
0
-2 and 2
y-intercepts
1
0
-3
4
2
0
0
5
-2
2
0
-2
-4
0
-4
symmetry
Even
Odd
Neither
Neither
Even
Neither
Odd
Even
Even
Neither
Odd
Neither
Neither
Odd
Even
Putting It Together (pages 22 to 31)
1.
Graph
Function
I
yes
II
yes
Domain
Range
Domain: {xεR }
Range: {yεR }
Domain: {xεR }
III
yes
Domain: {xεR - 1 ≤ x ≤ 4 }
Range: {yεR y ≤ 4 }
x-intercept
y-intercept
0
0
Odd, Even,
or Neither
odd
-4 and 0
0
neither
1
2
neither
Range: {yεR - 6 ≤ y ≤ 4 }
NSSAL
©2009
36
Draft
C. D. Pilmer
Graph
Function
IV
no
V
yes
VI
yes
Domain
Range
Domain: {xεR x ≥ −2 }
x-intercept
y-intercept
-2
-2 and 2
none
7
even
-4, -2, 0, 2,
and 4
0
odd
Domain: {xεR - 3 ≤ x ≤ 3 }
none
4
even
Domain: {xεR x ≤ 4 }
none
-4
Will not
classify.
Domain: {xεR - 15 ≤ x ≤ 15 }
0
-2, 0, and
2
Will not
classify.
Domain: {xεR - 3 ≤ x ≤ 3 }
-3
-3
neither
Domain: {xεR x ≥ −4 }
2
-1
neither
-2
none
Will not
classify.
2 and 7
4
neither
Range: {yεR }
Domain: {xεR }
Range: {yεR 1 ≤ y ≤ 7 }
Domain: {xεR - 4 ≤ x ≤ 4 }
Range: {yεR - 5 ≤ y ≤ 5 }
VII
yes
Odd, Even,
or Neither
Will not
classify.
Range: {4 }
VIII
no
Range: {yεR y ≤ −1 }
IX
no
Range: {yεR - 3 ≤ y ≤ 3 }
X
yes
Range: {yεR - 6 ≤ y ≤ 0 }
XI
yes
Range: {yεR y ≥ −3 }
XII
no
Domain: {- 2 }
XIII
yes
Domain: {xεR 0 ≤ x ≤ 8 }
Range: {yεR - 4 ≤ y ≤ 3 }
Range: {yεR - 2 ≤ y ≤ 4 }
2. (a)
(b)
(c)
(d)
Function
Domain: {tεR 0 ≤ t ≤ 5 }
Range: {dεR 0 ≤ d ≤ 4 }
t-intercept: 5
d-intercept: 2
Neither
3. (a)
(b)
(c)
(d)
Function
Domain: {tεR 0 ≤ t ≤ 4 }
t-intercept: none
Neither
NSSAL
©2009
Range: {cεR 30 ≤ c ≤ 120 }
c-intercept: 30
37
Draft
C. D. Pilmer