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Functions and Graphs
The Mathematics of Relations
Definition of a Relation
Relation
(A)
(B)
(C)
(1) 32 mpg
(2) 8 mpg
(3) 16 mpg
Domain and Range
• The values that make up the set of
independent values are the domain
• The values that make up the set of
dependent values are the range.
• State the domain and range from the 4
examples of relations given.
Quick Side Trip Into the Set
of Real Numbers
The Set of Real Numbers
Ponder
• To what set does the sum of a rational
and irrational number belong?
• How many irrational numbers can you
generate for each rational number using
this fact?
Properties of Real Numbers
• Transitive:
If a = b and b = c then a = c
• Identity:
a + 0 = a, a • 1 = a
• Commutative:
a + b = b + a, a • b = b • a
• Associative:
(a + b) + c = a + (b + c)
(a • b) • c = a • (b • c)
• Distributive:
a(b + c) = ab + ac
a(b - c) = ab - ac
Definition of Absolute Value
if a is positive
if a is negative
The Real Number Line
End of Side Trip Into the Set
of Real Numbers
Definition of a Relation
• A Relation maps a value from the
domain to the range. A Relation is a
set of ordered pairs.
• The most common types of relations in
algebra map subsets of real numbers to
other subsets of real numbers.
Example
Domain
Range
3
π
11
-2
1.618
2.718
Define the Set of Values that Make
Up the Domain and Range.
• The relation is the year and the cost of a
first class stamp.
• The relation is the weight of an animal
and the beats per minute of it’s heart.
• The relation is the time of the day and
the intensity of the sun light.
• The relation is a number and it’s square.
Definition of a Function
• If a relation has the additional
characteristic that each element of the
domain is mapped to one and only one
element of the range then we call the
relation a Function.
Definition of a Function
• If we think of the domain as the set of
boys and the range the set of girls, then
a function is a monogamous
relationship from the domain to the
range. Each boy gets to go out with one
and only one girl.
• But… It does not say anything about the
girls. They get to live in Utah.
FUNCTION CONCEPT
f
x
y
DOMAIN
RANGE
NOT A FUNCTION
R
x
y1
y2
DOMAIN
RANGE
FUNCTION CONCEPT
f
x1
y
x2
DOMAIN
RANGE
Examples
• Decide if the following relations are
functions.
X Y X Y
X Y
X Y
1
2
-5 7
1
1
1
2
-5 1
1
7
-1 2
-1 1
1
2
3
3
1
3
3
1
1
π
π
1 -1
5
Ponder
• Is 0 an even number?
• Is the empty set a function?
Ways to Represent a Function
• Symbolic
x,y y  2x
or
y  2x
• Numeric X Y
1
2
5 10
-1 -2
3
• Graphical
6
• Verbal
The cost is twice
the original
amount.
Example
• Penney’s is having a sale on coats. The
coat is marked down 37% from it’s
original price at the cash register.
• If you chose a coat that originally costs
$85.99, what will the sale price be? What
amount will you pay in total for the coat
(Assume you bought it in California.)
• Is this a function? What is the domain and
range? Give the symbolic form of the function.
If you chose a coat that costs $C, what will be
the amount $A that you pay for it?
Function Notation
The Symbolic Form
• A truly excellent notation. It is concise
and useful.
y  f x 
y  f x 
Name of the
function
• Output Value
• Member of the Range
• Dependent Variable
• Input Value
• Member of the Domain
• Independent Variable
These are all equivalent
names for the y.
These are all equivalent
names for the x.
Example of Function Notation
• The f notation
f x   x 1
f 2  2 1
Graphical Representation
• Graphical representation of functions
have the advantage of conveying lots of
information in a compact form. There
are many types and styles of graphs but
in algebra we concentrate on graphs in
the rectangular (Cartesian) coordinate
system.
Average National Price of Gasoline
Graphs and Functions
Range
Domain
CBR
Vertical Line Test for
Functions
• If a vertical line intersects a graph once
and only once for each element of the
domain, then the graph is a function.
Determine the Domain and
Range for Each Function
From Their Graph
Big Deal!
• A point is in the set of
ordered pairs that make up
the function if and only if
the point is on the graph of
the function.
Numeric
• Tables of points are the most common
way of representing a function
numerically
Verbal
• Describing the relation in words. We did
this with the opening examples.
Key Points
• Definition of a function
• Ways to represent a function
Symbolically
Graphically
Numerically
Verbally