A function is a rule that assigns each input exactly one output, so that each x-value relates to a single y-value. Functions can be represented numerically with ordered pairs, graphically with a set of points where no x-value appears twice, verbally through a word problem, and algebraically with an equation relating x and y. Checking for a single output for each input is important to determine if a relation represents a function.

1. 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

2. Functions Unit 4
Part 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.

3. 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)}

4. 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)}

5. What Will You Get?
If you combine cake mix, eggs and milk and put it in the oven, what will come out?
Cake
mix

6. What Will You Get
If you combine the ingredients again and put it in the oven, what will come out?
Cake
mix

7. 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}

8. 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}

9. Relations and Functions
• Relations and functions can also be
represented as relationships between two
sets of elements
Input Output Input Output
x-values y-values x-values y-values
Domain Range Domain Range
1 1 2
2
3 3 4
4
5 5 6
6
7 7 8
Relation/Function Relation/Not a Function

10. Relations and Functions
• Now you try. Determine whether each set
is a relation, a function, or both.
3
5 Amy Bob
6
10 Liz Joe
9
15 Sara Dan
21
a b
2 2
e c
4 4
i d
6 6
o f
8 8

11. Relations and Functions
• We will look at functions in four different
ways
1. Numerically; tables and ordered pairs
2. Graphically
3. Verbally
4. Algebraically

12. Functions-- Numerically
• For each x value, you can have one, and
only one, y value
• Check each table for repeating x’s
x y x y
4 2 4 4
2 2 2 2
0 0 0 0
-2 -2 2 -2
-4 -4 4 -2

13. 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)}

14. 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

15. 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

16. 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◦

17. Functions--Verbally

18. 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.
x y=x+1 y
-2 -2 + 1 -1
y=x+1 -1 -1 + 1 0
0 0+1 1
1 1+1 2
2 2+1 3
3 3+1 4

19. Functions