Relations and Functions

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Relations and Functions
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Which of these are related?

• 1. Body height and Age
• 2. Ethnicity and Intelligence
• 3. Diameter of tree and Age of tree
• 4. Thickness of guitar string and Pitch
• 5. Hours you work and Amount Earned
• 6. Brand of clothes and Coolness
• 7. Distance of Moon from Earth and Level of
  Tides
• 8. Speed of wind and Day of the week

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Relations

• Relation
• a mapping or pairing of input values with output
  values
• 5. The input is the of hours you work and the
  output is the amount you earn
• 3. The input is the diameter of the tree and
  the output is the age of the tree
• Domain Input values, or x (independent variable)
• Range Output values, or y (dependent variable)

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How do we represent relations?

• Ordered Pairs
• (Input, Output)
• (x, y)
• A (-4, 5)
• B (1, 2)
• C (3, -4)
• D (0, 3)
• E (-4, -3)

• Graph

A
D
B
E
C
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How do we represent relations cont.

• Table

• Mapping Diagram

Input
Output
-4
5
2
1
-4
3
3
0
-3
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Example

• (-3, 5), (-2, 1), (2, 7), (1, 5), (3, 6), (2, 1)
• Draw a mapping diagram for it.

-3
5
Input
-2
Output
1
2
7
1
6
3
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A special relationship

• Function
• a relation in which each input has exactly one
  output
• That means every number in the input column (x)
  gets paired with only one value in the output
  column (y)
• If even one input gets paired with two different
  outputs, then it is NOT a function!!

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Real World Example of a Function
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Is a vending machine a function?
16
13
26
4
25
Green Bean Milk
Choc. Milk
Lays Chips
Apple Milk
Lays Chips
Is it okay that both 26 and 25 are paired with
Lays Chips?
Would it be okay if 4 was paired with either
Apple Milk or Power Sweat?
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Example 1 Shipping a package
1
6
11
16
21
34
10
16
22
28
What is the domain?
1, 6, 11, 16, 21
What is the range?
10, 16, 22, 28, 34
Is this relation a function?
YES. IT IS A FUNCTION!!!
(Do all inputs go to only one output?)
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Example 1 Revisited

• In example 1, there were two variables.
• Weight of package
• Cost to ship it
• Which variable do you think depends on the other
  variable?
• The cost depends on the weight!!
• Independent Variable Input variable (Domain)
• Weight of the package
• Dependent Variable Output variable (Range)
• Cost to ship it

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How do we talk about functions?

• The dependent variable is a function of the
  independent variable
• Ex 1
• The cost to ship a package is a function of the
  weight of the package
• Vending machine
• The food item you receive is a function of the
  you type in

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Examples
Is this relation a function??
Input
Output
Yes, it is a function!!! Why? Because each input
goes to only one output
3
2
5
4
7
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Examples
Is this relation a function?
NO, it is not a function!! Why?? Because the
input (3) goes to two different places (36 and
35)
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Examples
Is this relation a function?
NO, it is not a function!! Why?? Because the
input (1) goes to two different outputs (16 and
-12)
-1
17
1
16
2
-12
3
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Examples
Is this relation a function?
YES, it is a function!! Why? Because each input
only goes to one output!!
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Which is a function?
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• Remember
• A function means each input (x) gets paired with
  only one output (y)!!

NOT A FUNCTION!!!
A FUNCTION!!!
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Vertical Line Test

• A relation is a function if and only if no
  vertical line intersects the graph at more than
  one point.

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Examples
Does the graph represent a function? Why or why
not?
It is a function because no vertical line
intersects it in more than one point!!
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Examples
Does the graph represent a function? Why or why
not?
It is not a function because a vertical line
intersects it in two points!!
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Does the graph represent a function??
NO, It is not a function because a vertical line
intersects it in two points!!
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Graph y -2x 1
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Linear Functions

• Any function of the form y mx b where m and b
  are constants (s)
• Ex y -2x 1 , y 5 0.5x
• Linear functions include
• Two different variables (usually x and y)
• Both variables to the power of one
• NO absolute values, square roots, etc.

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Function Notation

• Sometimes, we write y mx b as f(x) mx b
• By replacing y with f(x), we are using function
  notation
• Ex y 3x 4 is the same as f(x) 3x-4
• F(x) is pronounced f of x or the value of f
  at x
• You can also replace y with g(x), h(x), z(x),
  p(x), etc.
• The letter doesnt matter. They all are the same
  as y.
• If f(x) 4x 3, then f(7) 4(7) 3.
• So, f(7) means plug in the number 7 for x

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Linear or Not???

• 1.
• 2.
• 3.
• 4.
• 5.

Yes, y x 0
No, because the power on y is 2
Yes, g(x) 0x 10
No, because of the absolute value
Yes
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Evaluate.

• Find f(4) when f(x) -8x 13
• Find h(-12) when h(x) -x - 9

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Diver Propulsion Vehicles (DPV)
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• A diver using a DPV descends to a depth of 130
  feet. The pressure P (in atmospheres, atm) on
  the diver is given by P(d) 1 0.03d where d is
  the depth of the diver.
• Discuss a reasonable domain and range. Then
  graph the function.
• What is the pressure on the diver at a depth of
  33 feet?

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Homework

• P77/ 7-17, 21-26, 33-36, 42-43