3.2 Relations

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3.2 Relations And Functions
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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
This is a relation
(2,3), (-1,5), (4,-2), (9,9), (0,-6)
(2,3), (-1,5), (4,-2), (9,9), (0,-6)
(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
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Review

• A relation between two variables x and y is a set
  of ordered pairs
• An ordered pair consist of a x and y-coordinate
• 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

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Example 1

• What is the domain?
• 0, 1, 2, 3, 4, 5
• What is the range?
• -5, -4, -3, -2, -1, 0

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A relation assigns the xs with ys
1
2
2
4
3
6
4
8
10
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Domain (set of all xs)
Range (set of all ys)
This relation can be written (1,6), (2,2),
(3,4), (4,8), (5,10)
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A function is a relation that has each input
produce ONLY ONE output.
No x has more than one y assigned
This is a function ---it meets our conditions
The x value can only be assigned to one y
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Lets 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.
No x has more than one y assigned
This is a function ---it meets our conditions
Must use all the xs
The x value can only be assigned to one y
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1
2
2
4
3
6
4
8
10
5
2 was assigned both 4 and 10
Is the relation shown above a function?
NO
Why not???
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Example 2
4
5
0
9
1
Input
2
7
Output
THIS IS NOT A FUNCTION!!!
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Example 3

• Is this a function?
• Hint Look only at the x-coordinates

• YES

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Example 4

• 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 Choice Two

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

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.
There is only one price for each different item
on a certain date. The relation from items to
price makes it a function.
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Function Notation
We commonly call functions by letters. Because
function starts with f, it is a commonly used
letter to refer to functions.
This means the right hand side is a function
called f
The left side DOES NOT MEAN f times x like
brackets usually do, it simply tells us what is
on the right hand side.
This means the right hand side has the variable x
in it
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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.
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).
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Find f (-2).
This means to find the function f and instead of
having an x in it, put a -2 in it. So lets take
the function above and make brackets everywhere
the x was and in its place, put in a -2.
Dont forget order of operations---powers, then
multiplication, finally addition subtraction
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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) f(1.5) f(-4)
3
4
4
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For each function, evaluate f(0), f(1.5),
f(-4),
f(0) f(1.5) f(-4)
1
3
1
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For each function, evaluate f(0), f(1.5),
f(-4),
f(0) f(1.5) f(-4)
-5
1
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

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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
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Is the following function discrete or continuous?
What is the Domain? What is the Range?
continuous
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Is the following function discrete or continuous?
What is the Domain? What is the Range?
continuous
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Is the following function discrete or continuous?
What is the Domain? What is the Range?
discrete