POD

1. What is the longest distance between any two points in a crate with the following dimension:

Length 10 feet

Width 9 feet

Height 4 feet

Functions Unit 4Part 1

CC8.F.1 Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output.

What is a Relation? A rule that gives an output number for every

valid input number A set of ordered pairs for which all x and y

values are related in the same way. No special rules need apply.

The following are examples of relations: {(1,2), (1, 4), (1, 5), (1, 6), (1, -3)}

{(1,2), (2, 4), (3, 5), (2, 6), (1, -3)}

What is a Function? A rule of matching elements of two sets of

numbers in which an input value from the first set has only one output value in the second set.

Every value of x has a unique value of y.

function: {(1,2), (2, 4), (3, 5), (4, 6), (5,-3)}

What Will You Get?

Cake mix

If you combine cake mix, eggs and milk and put it in the oven, what will come out?

What Will You Get

Cake mix

If you combine the ingredients again and put it in the oven, what will come out?

Domain

• In a function, the possible values for x in the given situation.

• It is the set of values of the independent variable of a given function.

function:  {(1,2), (2, 4), (3, 5), (4, 6), (5,-3)}

Domain: {1, 2, 3, 4, 5}

Range

• In a function, the possible values for y in the given situation.

• It is the set of values of the dependent variable of a given function.

function:  {(1,2), (2, 4), (3, 5), (4, 6), (5,-3)}

Range: {2, 4, 5, 6, -3}

Relations and Functions

• Relations and functions can also be represented as relationships between two sets of elements

1

3

5

7

2

4

6

Relation/Function

1

3

5

7

2

4

6

8

Relation/Not a Function

Inputx-valuesDomain

Inputx-valuesDomain

Outputy-valuesRange

Outputy-valuesRange

Relations and Functions• Now you try. Determine whether each set

is a relation, a function, or both.3

6

9

21

5

10

15

a

e

i

o

2

4

6

8

b

c

d

f

Bob

Joe

Dan

Amy

Liz

Sara

2

4

6

8

Relations and Functions

• We will look at functions in four different ways

1. Numerically; tables and ordered pairs

2. Graphically

3. Verbally

4. Algebraically

Functions-- Numerically

• For each x value, you can have one, and only one, y value

• Check each table for repeating x’s

x y4 22 20 0-2 -2-4 -4

x y4 42 20 02 -24 -2

Functions-- Numerically

• For each x value, you can have one, and only one, y value

• Check each set of ordered pairs for repeating x’s

• {(4,4), (2,2), (0,0), (-2,-2), (-4,-4)}

• {(4,4), (2,2), (0,0), (2,-2), (4,-4)}

Functions-- Graphically

• For each x value, you can have one, and only one, y value

• Check that each x-coordinate is related to only one y-coordinate

Functions--Graphically

• For each x value, you can have one, and only one, y value

• Check that each x-coordinate is related to only one y-coordinate

Functions--Verbally

• It is a surprising biological fact that most crickets chirp at a rate that increases as the temperature increases. For the snowy tree cricket (Oecanthus fultoni), the relationship between temperature and chirp rate is so reliable that this type of cricket is called the thermometer cricket. We can estimate the temperature (in degrees Fahrenheit) by counting the number of times a snowy tree cricket chirps in 15 seconds and adding 40. For instance, if we count 20 chirps in 15 seconds, then a good estimate of the temperature is

20 + 40 = 60◦F

Functions--Verbally• The number of gallons of paint needed to paint a house

depends on the size of the house. A gallon of paint typically covers 250 square feet. Thus, the number of gallons of paint, is a function of the area to be painted.

• The amount of paint needed DEPENDS on the area to be painted.

• Area to be painted is the independent variable; x or input• Gallons of paint is the DEPENDent variable; y or output

Functions--Algebraically• For each x value, you can have one, and only

one, y value• Check that each x-value in your domain relates

to only one y-value in your range.

y = x + 1

x y = x + 1 y

-2 -2 + 1 -1

-1 -1 + 1 0

0 0 + 1 1

1 1 + 1 2

2 2 + 1 3

3 3 + 1 4

Functions