Functions
A relation is a set of ordered pairs of the form (x, y). The equation y = x + 4 describes
a relation. It relates the value of x to the value of y. A relation can be represented as an
equation, a graph, a table, a mapping diagram, or a list of ordered pairs.
A function is a special kind of relation in which each input is mapped to exactly one
output (for each input, there is exactly one output).
The output (y) is dependent on the input (x)
Function:
(1, 6) (3, 8) (5, 10)
Function
(1, 6) (3,8) (3, 9)
Not a function,
(the input 3 has two outputs; 8 and 9)
A relation can have a variety of representations:
{(1, 4), (2, 3), (6, 5)} LIST
Example:
x
1
2
6
Table
y
4
3
5
1
2
6
4
3
5
Mapping
Graph
A function consists of:
 DOMAIN (x)- The set of all possible inputs of the function
The domain for the above relation/function is {1, 2, 6}
 RANGE (y) - The set of all possible outputs of the function
The range for the above relation/function is {4, 3, 5}
Input-Output Table
Domain
Independent Variable
II.
Input
Output
x
1
2
6
y
4
3
5
Range
Dependent Variable
Identify the domain and range of a function
The input-output table shows temperatures over
various increments of time.
To identify the domain, look at the set of inputs: 0, 2, 4, 6
To identify the range, look at the set of outputs: 24, 27, 30, 33
Try It!!! Identify the domain and range
Domain:
Range:
III.
Identify a Function
A.
A. The pairing is a function because
each input is paired with exactly
one output
B.
B. The pairing is NOT a function because
the input “2” is paired with the output 2
and the output 4. It can only be paired
with one output not two
Try It!!!! Tell whether the pairing is a function and why.
2
3.
IV.
4.
FUNCTIONS:
A function may be represented using a “rule” that relates one variable to another.
V.
Make a Table for a FUNCTION
The domain of the function rule 𝑦 = 3𝑥 is: {𝟎, 𝟏, 𝟐, 𝟑}.
1 Make a table
Input
Domain
Independent
Variable
x
Rule
𝑦 = 3𝑥
2. Use the input (domain), to calculate the output,
(range)
Input
Domain
Independent
Variable
Output
Range
Dependent
Variable
x
0
1
2
3
y
Rule
𝑦 = 3𝑥
𝑦 = 3(0)
𝑦 = 3(1)
𝑦 = 3(2)
𝑦 = 3(3)
Output
Range
Dependent
Variable
y
0
3
6
9
The range of the function 𝑦 = 3𝑥 is {𝟎, 𝟑, 𝟔, 𝟗}
VI.
Write a function rule
Try It!!!! Write a rule for the function
Identify the domain and range
Function Rule:
Domain:
Range:
Sometimes, machine diagrams are used to represent functions.
In the function machine below, the inputs are labeled x and the outputs are labeled y.
The function is represented by the expression 2x + 5.
a. If x = 7 is used as an input, what is the output? 2(7) + 5 = 19
b. If x = -2 is used as an input, what is the output? 2(−2) + 5 = 1
You Try It!
1
c. If x = is used as an input, what is the output?
2
d. Is there any limit to the number of input values that can be used with this expression?
Explain
Each of the functions that you have seen has a finite number of ordered pairs.
There are functions that have an infinite number of ordered pairs.
Describe any difficulties that may exist trying to represent a function with an infinite
number of ordered pairs using the four representations of functions that have been
described thus far. (Table, mapping, graphing, listing)