Transcript Functions

FUNCTIONS & RELATIONS
FUNCTIONS & RELATIONS
yy axis
The Coordinate Plane
10
9
8
QUADRANT
II
(– , +)
(x, y) = ordered pair
QUADRANT
I
(+ , +)
7
6
5
Abscissa – The first coordinate of an
ordered pair of real numbers that is
assigned to a point on the
coordinate plane. (x – coordinate)
4
3
The Origin: (0, 0)
2
1
–10 –9
–8
–7
–6
–5
–4
–3
–2
–1
–1
1
2
3
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5
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–2
–3
Ordinate – The second coordinate of
an ordered pair of real numbers that
is assigned to a point on the
coordinate plane. (y – coordinate)
–4
QUADRANT
III
(– , –)
–5
–6
–7
–8
–9
–10
QUADRANT
IV
(+ , –)
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8
9
10
xx axis
Identify which point is…
y
10
Located in the
Fourth Quadrant.
9
 Located on the
x – axis.
5
 Located on the
y – axis.
 Located in the
Second Quadrant.
 Located at (6, –6).
8
•G
7
6
•C
4
•A
3
•B
2
–10 –9
•
D
–8
–7
1
–6
–5
–4
–3
–2
–1
–1
–2
–3
–4
•F
–5
–6
–7
–8
–9
–10
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•E
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10
x
Relation:
A relation is any set of ordered pairs.
Consider the set {(0, 2), (– 4, 3), (–3, –2), (2, –1)}.
• The set of all first coordinates of the ordered pairs is called the domain
of the relation and is often named D.
• The set of all second coordinates of the ordered pairs is called the range
of the relation and is often called R.
Domain and Range
• Therefore, in the relation {(0, 2), (–4, 3), (–3, –2), (2,–1)}…
D = {0, –4, –3, 2}
R = {2, 3, –2, –1}
The domain is the set of independent
variable, x’s, and the range is the set of
dependent variables, y’s.
Relation:
There are many ways to represent a relation.
1) Set Notation using braces:
{(0, 2), (– 4, 3), (–3, –2), (2, –1)}.
2) Table:
3) Mapping Diagram:
Functions:
If no two ordered pairs in a relation have the
same first coordinate, the relation is called a
function.
A function is the pairing between two sets of numbers in
which every element in the first set is paired with exactly one
element of the second set. In other words…
In a function, for every x there is only one y.
Functions:
Real Life Examples:
A relation can be a relationship between sets of information.
For example, consider the set of all of the people in your Algebra class
and the set of their heights is a relation. The pairing of a person and his
or her height is a relation.
In relations and functions, the pairs of names and height are ordered,
which means one comes first and the other comes second. These two
sets could be ordered with the person first {student, height} or the
person last {height, student}.
While all functions are relations, since they pair information, not all
relations are functions.
Real Life Examples:
In the student/height example, the relation {student, height} is a function
because every person has only one height at any given point in time.
The relation {height, student} is not a function because for every height
there might be many students that are that tall.
Other examples of relations that are functions:
{person, social security number}
{dog, tail}
{HMS student, homebase}
Other examples of relations that are not functions:
{person, telephone number}
{HMS student, club or team}
{dog, feet}
The Vertical Line Test
Given the graph of a relation, if you can draw a vertical line that crosses the
graph at more than one point, then the relation is not a function.
This relation IS as function!
The Vertical Line Test
Given the graph of a relation, if you can draw a vertical line that crosses the
graph at more than one point, then the relation is not a function.
This relation IS NOT a function!
Function Notation
A rule for a relation is said to define the relation. Given a
rule and a domain (the independent variable(s) x) for a
relation or function, it is possible to determine all the
ordered pairs that form the relation, which will determine
the range (the dependent variable(s) y) of the relation.
EXAMPLE: Find the range, R, of the relation defined by the
rule 2x + 1, if the domain, D, = {0, 1, 2, – 1, –2}
Function Notation
EXAMPLE: Find the range, R, of the relation defined by the
rule 2x + 1, if the domain, D, = {0, 1, 2, – 1, –2}
An expanded “T” table can be used to organize the data:
Therefore, R = {1, 3, 5, – 1, –5}
Function Notation
A function is named by a single letter, such as f, F, or g.
For example, the function defined by the rule 2x + 1 may be
called f and can specified in two ways:
Arrow Notation:
f:x  2x + 1
“The function f that pairs a number x with the number 2x + 1”
Function Notation: f(x) = 2x + 1
“f of x” is used to denote the specific value of the function
that is paired with the number x.
Function Notation
Example #1: State the domain and range of the function
{(0, 2), (2, 4), (4, 6), (6, 8)}.
{(0, 2), (2, 4), (4, 6), (6, 8)}.
D = {0, 2, 4, 6 }
R = { 2, 4, 6, 8 }
Function Notation
Example #2: Given D = {–2, –1, 0, 1, 2 }, determine range
of the function f(x) = x – 1.
R = {–3, –2, –1, 0, 1}
* * Note: the order of the range should match
the order of the domain. There is a one-to-one
correspondence between each domain value and
each range value.
Function Notation
Example #3: Find each of the following for the function f(x) = 2x – 4
Function Notation
Example #3: Find each of the following for the function f(x) = 2x – 4
Function Notation
Example #4:
Find each of the following for the function g(x) = 3x2 + 2x + 4
1) g(0) = 4
2) g(1) = 9
3) g(–1) = 5
Function Notation
Example #4:
Find each of the following for the function g(x) = 3x2 + 2x + 4
4) g
1
2
5) g
1
−
2
6) g
3
4
=5 =
2
−
3
=
3
3
4
=4
23
4
=
15
4