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Relations and Functions
 Analyze and graph relations.
 Find functional values.
1) ordered pair
2) Cartesian Coordinate
3) plane
4) quadrant
5) relation
6) domain
7) range
8) function
9) mapping
10) one-to-one function
11) vertical line test
12) independent variable
13) dependent variable
14) functional notation
Relations and Functions
This table shows the average lifetime
and maximum lifetime for some animals.
The data can also be represented as
ordered pairs.
The ordered pairs for the data are:
Average
Lifetime
(years)
Maximum
Lifetime
(years)
Cat
12
28
Cow
15
30
Deer
8
20
Dog
12
20
Horse
20
50
Animal
(12, 28), (15, 30), (8, 20),
(12, 20), and (20, 50)
The first number in each ordered pair
is the average lifetime, and the second
number is the maximum lifetime.
(20, 50)
average
lifetime
maximum
lifetime
Relations and Functions
You can graph the ordered pairs below
on a coordinate system with two axes.
Animal Lifetimes
(12, 20), and (20, 50)
50
Remember, each point in the coordinate
plane can be named by exactly one
ordered pair and that every ordered pair
names exactly one point in the coordinate
plane.
Maximum Lifetime
(12, 28), (15, 30), (8, 20),
60
y
40
30
20
10
The graph of this data (animal lifetimes)
lies in only one part of the Cartesian
coordinate plane – the part with all
positive numbers.
0
x
0
5
10
15
20
Average Lifetime
25
30
Relations and Functions
In general, any ordered pair in the coordinate plane can be written in the form (x, y)
A relation is a set of ordered pairs, such as the one for the longevity of animals.
The domain of a relation is the set of all first coordinates (x-coordinates) from the
ordered pairs.
The range of a relation is the set of all second coordinates (y-coordinates) from the
ordered pairs.
The graph of a relation is the set of points in the coordinate plane corresponding to the
ordered pairs in the relation.
•
•
•
•
Domain
Input
x
Independent variable
SYNONYMS FOR “DOMAIN”
•
•
•
•
Range
Output
y
Dependent variable
SYNONYMS FOR “RANGE”
Relations and Functions
A function is a special type of relation in which each element of the domain is paired
with ___________
exactly one element in the range.
A mapping shows how each member of the domain is paired with each member in
the range.
Functions
 3,1, 0,2, 2,4
Domain
Range
-3
1
0
2
2
4
one-to-one function
Relations and Functions
A function is a special type of relation in which each element of the domain is paired
with ___________
exactly one element in the range.
A mapping shows how each member of the domain is paired with each member in
the range.
Functions
 1,5, 1,3, 4,5
Range
Domain
-1
5
1
3
4
function,
not one-to-one
Relations and Functions
A function is a special type of relation in which each element of the domain is paired
with ___________
exactly one element in the range.
A mapping shows how each member of the domain is paired with each member in
the range.
Functions
5,6,  3,0, 1,1,  3,6
Domain
Range
5
6
-3
0
1
1
not a function
Relations and Functions
The Cartesian coordinate system is composed of the x-axis (horizontal),
and the y-axis (vertical), which meet at the origin (0, 0) and divide the plane into
four quadrants.
You can tell which quadrant a point is in by looking at the sign of each coordinate of
the point.
5
Quadrant
OriginI
( +, (0,
+ )0)
Quadrant II
( --, + )
0
-5
0
5
Quadrant III
( --, -- )
Quadrant IV
( +, -- )
-5
The points on the two axes do not lie in any quadrant.
Relations and Functions
State the domain and range of the relation shown
in the graph. Is the relation a function?
y
(-4,3)
(2,3)
The relation is:
{ (-4, 3), (-1, 2), (0, -4), (2, 3), (3, -3) }
x
The domain is:
{ -4, -1, 0, 2, 3 }
(-1,-2)
The range is:
{ -4, -3, -2, 3 }
(0,-4)
Each member of the domain is paired with exactly one member of the range,
so this relation is a function.
(3,-3)
Relations and Functions
You can use the vertical line test to determine whether a relation is a function.
Vertical Line Test
If no vertical line intersects a
graph in more than one point,
the graph represents a function.
y
x
Relations and Functions
You can use the vertical line test to determine whether a relation is a function.
Vertical Line Test
If no vertical line intersects a
graph in more than one point,
the graph represents a function.
If some vertical line intercepts a
graph in two or more points, the
graph does not represent a function.
y
y
x
x
Relations and Functions
The table shows the population of
Indiana over the last several decades.
We can graph this data to determine
if it represents a function.
Year
Population
(millions)
1950
3.9
1960
4.7
1970
5.2
1980
5.5
1990
5.5
2000
6.1
Population of Indiana
8
7
Population
(millions)
6
Use the vertical
line test.
5
4
3
Notice that no vertical line can be drawn
that contains more than one of the data
points.
2
1
0
‘50
‘60
‘70
‘80
Year
‘90
‘00
0
7
Therefore, this relation is a function!
Relations and Functions
Graph the relation y  2 x  1
2) Graph the ordered pairs.
1) Make a table of values.
y
7
x
y
-1
-1
0
6
5
4
3
2
1
1
1
3
x
0
-1
2
5
-2
-3
-5
-4
-3
-2
-1
1
2
3
4
5
0
3) Find the domain and range.
Domain is all real numbers.
4) Determine whether the relation is a function.
The graph passes the vertical line test.
Range is all real numbers.
For every x value there is exactly one y value,
so the equation y = 2x + 1 represents a function.
Relations and Functions
Graph the relation x  y 2  2
2) Graph the ordered pairs.
y
1) Make a table of values.
7
6
x
5
y
4
2
-2
3
2
-1
-1
1
x
0
-2
0
-1
-2
-1
1
-3
-5
-4
-3
-2
-1
1
2
3
4
5
0
2
2
3) Find the domain and range.
Domain is all real numbers,
greater than or equal to -2.
Range is all real numbers.
4) Determine whether the relation is a function.
The graph does not pass the vertical line test.
For every x value (except x = -2),
there are TWO y values,
so the equation x = y2 – 2
DOES NOT represent a function.
Relations and Functions
When an equation represents a function, the variable (usually
up the domain is called the independent variable.
x) whose values make
The other variable (usually y) whose values make up the range is called the
dependent variable because its values depend on x.
Equations that represent functions are often written in function notation.
The equation y = 2x + 1 can be written as f(x) = 2x + 1.
y , and is read “f
The symbol f(x) replaces the __
of x”
The f is just the name of the function. It is NOT a variable that is multiplied by x.
Relations and Functions
Suppose you want to find the value in the range that corresponds to the element
4 in the domain of the function.
f(x) = 2x + 1
This is written as
The value
f(4)
f(4)
and is read
“f
of 4.”
is found by substituting 4 for each x in the equation.
Therefore, if
Then
f(x) = 2x + 1
f(4) = 2(4) + 1
f(4) = 8 + 1
f(4) = 9
NOTE: Letters other than f can be used to represent a function.
EXAMPLE:
g(x) = 2x + 1
Relations and Functions
Given:
f(x) = x2 + 2
and
g(x) = 0.5x2 – 5x + 3.5
Find each value.
f(-3)
g(2.8)
f(x) = x2 + 2
g(x) = 0.5x2 – 5x + 3.5
f(-3) = (-3)2 + 2
g(2.8) = 0.5(2.8)2 – 5(2.8) + 3.5
f(-3) = 9 + 2
g(2.8) = 3.92 – 14 + 3.5
f(-3) = 11
g(2.8) = – 6.58
Relations and Functions
Given:
f(x) = x2 + 2
Find the value.
f(3z)
f(x) = x2 + 2
f( 3z ) = (3z) 2 + 2
f(3z) = 9z2 + 2
Relations and Functions