Functions

Simplify
1. 3 + 4(5 + 6) - 25(4 - 3(2))
2. 3x^2 + 4x - 15 at x = 2

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

Cat 12 28
Cow 15 30
Deer 8 20
Dog 12 20
Horse 20 50
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:
(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

y
x
30
20
40
60
5 25
10
50
30
0
0 10 15
20
Average Lifetime
Maximum
Lifetime
(12, 28), (15, 30), (8, 20),
(12, 20), and
(20, 50)
You can graph the ordered pairs below
on a coordinate system with two
axes.
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.
The graph of this data (animal lifetimes)
lies in only one part of the Cartesian
coordinate plane – the part with all
positive numbers.

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.
0 5
-5
0
5
-5
Origi
n
(0, 0)
Quadrant I
( +, + )
Quadrant II
( --, + )
Quadrant III
( --, -- )
Quadrant IV
( +, -- )
The points on the two axes do not lie in any quadrant.

In general, any ordered pair in the coordinate
plane can be written in the form (x, y).
What is a RELATION?
A relation is a set of ordered pairs.
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.

Given the relation:
{(2, -6), (1, 4), (2, 4), (0,0), (1, -6), (3, 0)}
State the domain:
D: {0,1, 2, 3}
State the range:
R: {-6, 0, 4}
Note: { } are the symbol for "set".
When writing the domain and
range, do not repeat values.

{(2, –3), (4, 6), (3, –1), (6, 6), (2, 3)}
domain: {2, 3, 4, 6}
range: {–3, –1, 3, 6}
State the domain and range of the following
relation.

y
x
(-4,3) (2,3)
(-1,-2)
(0,-4)
(3,-3)
State the domain and range of the relation
shown in the graph.
The relation is:
{ (-4, 3), (-1, 2), (0, -
4),
(2, 3), (3, -3) }
The domain is:
{ -4, -1, 0, 2, 3 }
The range is:
{ -4, -3, -2, 3 }

• Relations can be written in several
ways: ordered pairs, table, graph, or
mapping.
• We have already seen relations
represented as ordered pairs.

Table
{(3, 4), (7, 2),
(0, -1), (-2, 2),
(-5, 0), (3, 3)}
x y
3 4
7 2
0 -1
-2 2
-5 0
3 3

Mapping
• Create two ovals with the domain on
the left and the range on the right.
• Elements are not repeated.
• Connect elements of the domain with
the corresponding elements in the
range by drawing an arrow.

Mapping
{(2, -6), (1, 4), (2, 4), (0, 0), (1, -6), (3, 0)}
2
1
0
3
-6
4
0

FUNCTIONS
Objective: To recognize whether
a relation is a function or not.

What is a
FUNCTION?

{( − 3,1),( 0,2), ( 2,4)
Domain Range
3 1
0 2
2 4
function
ONE-TO-ONE
CORRESPONDENCE
A function is a special type of relation in which each element of the domain is
paired with exactly one element in the range.
Furthermore, a set of ordered pairs is a function if no two ordered pairs
have equal abscissas.
A mapping shows how each member of the domain is paired with each member
in the range.
Functions

{( −1,5), (1,3), ( 4,5)}
Domain Range
-1
1
5
3
4
function,
not one-to-one
MANY-TO-ONE
CORRESPONDENCE
paired with _exactly one element in the range.
in the range.
Functions

paired element in the range.
with exactly one
{(5,6), ( − 3,0),(1,1),( − 3,6)}
Domain Range
5
6
-3 0
1 1
not a function
ONE-TO-MANY
CORRESPONDENCE
in the range.
Functions

y
x
(-4,3) (2,3)
(-1,-2)
(0,-4)
(3,-3)
State the domain and range of the relation
shown in the graph. Is the relation a
function?
The relation is:
{ (-4, 3), (-1, 2), (0, -4), (2, 3), (3, -3) }
The domain is:
{ -4, -1, 0, 2, 3 }
The range is:
{ -4, -3, -2, 3 }
Each member of the domain is paired
with Exactly one member of the range, so
this relation is a function.

Function Not a Function
(4,12)
(5,15)
(6,18)
(7,21)
(8,24)
(4,12)
(4,15)
(5,18)
(5,21)
(6,24)

10
3
4
7
5
2
3
4
8
10
3
5
7
2
2
3
4
7
5

-3
-2
-1
0
1
-6
-1
-0
3
15
-3
-2
-1
0
1
-6
-1
-0
3
15

X Y
1 2
2 4
3 6
4 8
5 10
6 12
X Y
1 2
2 4
1 5
3 8
4 4
5 10

Function
Not a Function
X -3 0 3 8 -10
Y 6 8 20 4 8
X -2 0 -2 7 -8
Y 6 8 20 4 8

You can use the vertical line test to determine whether a relation is a function.
Vertical Line Test
y
x
y
x
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.

Year
Population
(millions)
1950 3.9
1960 4.7
1970 5.2
1980 5.5
1990 5.5
2000 6.1
The table shows the population of
Indiana over the last several decades.
We can graph this data to determine
if it represents a function.
7
8
7
6
5
4
3
2
1
0 ‘50 ‘60 ‘70
‘80 ‘90
‘00
0
Population
(millions)
Year
Population of Indiana
Use the vertical
line test.
Notice that no vertical line can be drawn
that contains more than one of the data
points.
Therefore
,
this relation is a function!
Therefore, this relation is a function!

Is this relation a function?
{(1,3), (2,3), (3,3)}
1. Yes
2. No
Answer Now

Vertical Line Test (pencil
test)
If any vertical line passes
through more than one point
of the graph, thenthat relation is
not a function.
Are these functions?
FUNCTION! FUNCTION! NOPE!

Vertical Line Test
NO!
FUNCTION!
FUNCTION!
NO!

When an equation represents a function, the
variable (usually x) whose values make up the
domain is called the independent variable.
The other variable (usually y) whose values
make up the range is called the dependent
variable because its values depend on x.

Function Notation
• When we know that a relation is a
function, the “y” in the equation can
be replaced with f(x).
• f(x) is simply a notation to designate a
function. It is pronounced ‘f’ of ‘x’.
• The ‘f’ names the function, the ‘x’ tells
the variable that is being used.
NOTE: Letters other than f can be used to
represent a function.
EXAMPLE: g(x) = 2x + 1

Function Notation
Output
Input
Name of
Function
y = f
(x)

Given f(x) = 3x - 2, find:
1) f(3) =
7
3(3)-2 7
3(-2)-2
-2 -8
3
2) f(-2) = -
8

Given h(z) = z2 - 4z + 9, find
h(- 3)
(-3)2-4(-3)+9
-3 30
9 + 12 + 9
h(-3) =

Given: f(x) = x2 + 2 and g(x) = 0.5x2 – 5x +
3.5
Find each value.

3.5
Find each value.
f(-3)

3.5
Find each value.
f(-3)
f(x) = x2 + 2

3.5
Find each value.
f(-3)
f(x) = x2 + 2
f(-3) = (-3)2 + 2

3.5
Find each value.
f(-3)
f(x) = x2 + 2
f(-3) = (-3)2 + 2
f(-3) = 9 + 2

3.5
Find each value.
f(-3)
f(x) = x2 + 2
f(-3) = (-3)2 + 2
f(-3) = 9 + 2
f(-3) = 11

3.5
Find each value.
f(-3)
f(x) = x2 + 2
f(-3) = (-3)2 + 2
f(-3) = 9 + 2
f(-3) = 11
g(2.8)

3.5
Find each value.
f(-3)
f(x) = x2 + 2
f(-3) = (-3)2 + 2
f(-3) = 9 + 2
f(-3) = 11
g(2.8)
g(x) = 0.5x2 – 5x + 3.5

3.5
Find each value.
f(-3)
f(x) = x2 + 2
f(-3) = (-3)2 + 2
f(-3) = 9 + 2
f(-3) = 11
g(2.8)
g(x) = 0.5x2 – 5x + 3.5
g(2.8) = 0.5(2.8)2 – 5(2.8) + 3.5

3.5
Find each value.
f(-3)
f(x) = x2 + 2
f(-3) = (-3)2 + 2
f(-3) = 9 + 2
f(-3) = 11
g(2.8)
g(x) = 0.5x2 – 5x + 3.5
g(2.8) = 0.5(2.8)2 – 5(2.8) + 3.5
g(2.8) = 3.92 – 14 + 3.5

3.5
Find each value.
f(-3)
f(x) = x2 + 2
f(-3) = (-3)2 + 2
f(-3) = 9 + 2
f(-3) = 11
g(2.8)
g(x) = 0.5x2 – 5x + 3.5
g(2.8) = 0.5(2.8)2 – 5(2.8) + 3.5
g(2.8) = 3.92 – 14 + 3.5
g(2.8) = – 6.58

Given g(x) = x2 – 2, find g(4)
Answer Now
1.
2
2.
6
3.
14
4.

Given f(x) = 2x + 1, find
f(-8)
Answer Now
1. -
40
2. -
15
3. -
8
4. 4

Value of a
Function
If g(s) = 2s + 3, find g(-2).
g(-2) = 2(-2) + 3
=-4 + 3
= -1
g(-2) = -1

Value of a
Function
If h(x) = x2 - x + 7, find
h(2c). h(2c) = (2c)2 –
(2c) + 7
= 4c2 - 2c + 7

Value of a
Function
If f(k) = k2 - 3, find f(a
- 1)
f(a - 1)=(a - 1)2 - 3
(Remember FOIL?!)
=(a-1)(a-1) - 3
= a2 - a - a + 1
- 3

Graph the relation y = 2x
+1
1) Make a table of values.
x y
-1 -1
0 1
1 3
2 5
3) Find the domain and range.
Domain is all real numbers.
Range is all real numbers.
2) Graph the ordered
pairs.
0
y
0 x
-1
2
4
6
-2
3
1
-3
-5 -4 -3 -2
-1 1 2
3 4 5
5
7
4) Determine whether the relation is a
function. The graph passes the vertical line
test.
F
s
r every x valu
o the equation
e there is exac
y = 2x + 1 repr
tly one y value
esents a
,
on
Foor every x value there is exactly one y value,
so the equation y = 2x + 1 represents a function.

Graph the relation x = y2 −
2
1) Make a table of values.
x y
2 -2
-1 -1
-2 0
-1 1
2 2
3) Find the domain and range.
Domain is all real numbers,
greater than or equal to -2.
Range is all real numbers.
2) Graph the ordered
pairs. y
0 x
-3
-1
2
4
6
-2
3
1
-5 -4 -3 -2
-1 1 2
3 4 5
5
7
0
4) Determine whether the relation is a
function.
The graph does not pass the vertical line test.
For ever
there
so t
DOES N
y x value (exce
are TWO y va
he equation x =
OT represent
pt x = -2),
lues,
y2 – 2
a function.
For every x value (except x = -
2), there are TWO y
values,
so the equation x = y2 – 2
DOES NOT represent a

Graphs of a Function
Vertical Line Test:
If a vertical line is passed
over the graphandit intersects
the graph in exactly one
point, the graph
represents a function.

x
y
x
y
Does the graph represent
a
function? Name the domain
and range.
Yes
D: all real
numbers R: all
real numbers
Yes
D: all real
numbers R: y ≥ -
6

x
y
x
y
Does the graph
represent a
and range.
No
D: x ≥ 1/2
R: all real
numbers
No
D: all real
numbers R: all
real numbers

Does the graph represent
a
and range.
Yes
D: all real
numbers R: y ≥ -
6
No
D: x = 2
R: all real
numbers
x
y
x
y

Functions

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