Download PPT 3.2 Relations and Functions

3.2 Relations
And
Functions
A relation is a set of ordered pairs.
The domain is the set of all x values in the relation
domain = {-1,0,2,4,9}
These are the x values written in a set from smallest to largest
{(2,3), (-1,5), (4,-2), (9,9), (0,-6)}
These are the y values written in a set from smallest to largest
range = {-6,-2,3,5,9}
The range is the set of all y values in the relation
This is a
relation
Review
• A relation between two variables x and y
is a set of ordered pairs
• An ordered pair consist of a x and ycoordinate
– A relation may be viewed as ordered pairs,
mapping design, table, equation, or written in
sentences
• x-values are inputs, domain, independent
variable
• y-values are outputs, range, dependent
variable
Example 1
{(0, 5), (1, 4), (2, 3), (3, 2), (4, 1), (5, 0)}
•What is the domain?
{0, 1, 2, 3, 4, 5}
What is the range?
{-5, -4, -3, -2, -1, 0}
A relation assigns the x’s with y’s
1
2
3
4
2
4
6
5
8
10
Domain (set of all x’s)
Range (set of all y’s)
This relation can be written {(1,6), (2,2), (3,4), (4,8), (5,10)}
A function is a relation that has each input produce ONLY
ONE output.
1
2
3
4
5
2
4
6
8
10
Set A is the domain
Set B is the range
This is a function
---it meets our
conditions
The x value can only be assigned to one y
Let’s look at another relation and decide if it is a function.
The second condition says each x can have only one y, but it CAN
be the same y as another x gets assigned to.
1
2
3
4
5
2
4
6
8
10
Set A is the domain
This is a function
---it meets our
conditions
Set B is the range
Must use all the x’s
The x value can only be assigned to one y
1
2
3
4
5
2
4
6
8
10
2 was assigned both 4 and 10
Is the relation shown above a function?
NO
Why not???
Example 2
4
Input
Output
–5
0
–2
9
–1
7
THIS IS NOT A FUNCTION!!!
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Example 3
{(0, 5), (1, 4), (2, 3), (3, 2), (4, 1), (5, 0)}
•Is this a function?
•Hint: Look only at the x-coordinates
YES
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Example 4
{(–1, 7), (1, 0), (2, 3), (0, 8), (0, 5), (–2, 1)}
•Is this a function?
•Hint: Look only at the x-coordinates
NO
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Example 5
Which mapping represents a function?
Choice One
3
1
0
Choice Two
2
–1
3
–1
2
3
2
3
–2
0
Choice 1
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Example 6
Which mapping represents a function?
A.
B.
B
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Example 7
Which situation represents a function?
a. The items in a store to their prices on a
certain date
b. Types of fruits to their colors
There is only one price for each
different item on a certain date. The
relation from items to price makes it a
function.
A fruit, such as an apple, from the
domain would be associated with
more than one color, such as red and
green. The relation from types of fruits
to their colors is not a function.
We commonly call functions by letters. Because function
starts with f, it is a commonly used letter to refer to
functions.
f  x   2 x  3x  6
2
This means
the right
hand side is
a function
called f
This means
the right hand
side has the
variable x in it
The left side DOES NOT
MEAN f times x like
brackets usually do, it
simply tells us what is on
the right hand side.
Remember---this tells you what
is on the right hand side---it is
not something you work. It says
that the right hand side is the
function f and it has x in it.
f  x   2 x  3x  6
2
f 2   22   32   6
2
f 2  24  32  6  8  6  6  8
So we have a function called f that has the variable x in it.
Using function notation we could then ask the following:
Find f (2).
Find f (-2).
f  x   2 x  3x  6
2
f  2  2 2  3 2  6
2
f  2  24  3 2  6  8  6  6  20
This means to find the function f and instead of having an x
in it, put a -2 in it. So let’s take the function above and make
brackets everywhere the x was and in its place, put in a -2.
Don’t forget order of operations---powers, then
multiplication, finally addition & subtraction
Function Notation
Given g(x) = x2 – 3, find g(-2) .
g(-2) = x2 – 3
g(-2) = (-2)2 – 3
g(-2) = 1
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For each function, evaluate f(0), f(1.5), f(-4),
f(0) = 3
f(1.5) = 4
f(-4) = 4
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For each function, evaluate f(0), f(1.5), f(-4),
f(0) = 1
f(1.5) =3
f(-4) =1
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For each function, evaluate f(0), f(1.5), f(-4),
f(0) = -5
f(1.5) =1
f(-4) =1
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Vertical Line Test
•Vertical Line Test: Tells you if a relation is a
function when a vertical line drawn through its
graph, passes through only one point.
Take a pencil and move it from left to right
(–x to x); if it crosses more than one point, it is not
a function
Vertical Line Test
Would this
graph be a
function?
YES
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Vertical Line Test
Would this
graph be a
function?
NO
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Is the following function discrete or continuous?
What is the Domain? What is the Range?
Discrete



-7, 1, 5, 7, 8, 10



1, 0, -7, 5, 2, 8
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Is the following function discrete or continuous?
What is the Domain? What is the Range?
continuous



8,8




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6,6

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Is the following function discrete or continuous?
What is the Domain? What is the Range?
continuous






0,45

10,70
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
27
Is the following function discrete or continuous?
What is the Domain? What is the Range?
discrete



-7, -5, -3, -1, 1, 3, 5, 7

2, 3, 4, 5, 7
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