1. Relations & Functions

2. What is a relation?
A relation is a set of ordered pairs (x,y) where x is the input value and y is the output value. The domain is all the possible inputs of a relation, and the range is all the possible outputs of a relation.
Below is an example of a relation expressed as a set of ordered pairs
{ (1,4), (2,8), (3,12), (4,16) }
The domain is { 1, 2, 3, 4 } The range is { 4, 8, 12, 16)
A relation can also be expressed as a mapping diagram, a chart, and a graph.

3. What is a function?
A function is a type of relation which there is only one output value for each input value (An x value can only be paired with only one y. An x or input value should never be repeated!)
Determine whether or not each relation is a function
{ (5,10), (6,20), (6,23), (7,35)}
{ (1,5), (2,3), (3,2), (4,1), (5,0) }
{ (57,9), (68,2), (93,9), (99,2) }
{ (6,3), (8,2), (9,0), (11,1), (11,2) }

4. The Vertical Line Test
If the graph of a relation is a function, it will pass the vertical line test. The vertical line test states that a relation is a function if and only if a vertical line does not pass through more than one point of the graph.

5. Modeling Using Function Notation
X, which is the input value is also called the independent variable. Y which is the output value is also called the dependent variable. In more mathematical terms, the dependent variable y is a function of the independent variable x. The function notation is denoted by F(x).
Example 1 – Amanda babysits and charges $5 per hour. Write an equation in function notation.
What is the dependent variable?
What is the independent variable?
Time worked in hrs. 1 2 3 4
Amount earned in
Dollars $ 5

6. Function notation f(x) = 5x
Time worked in hrs. 1 2 3 4
Amount earned in
Dollars $ 5
The amount of money Amanda earns is dependent upon the number of hours that she works, so the amount of money earned is the dependent variable and the # of hours worked is the independent variable. The amount earned is equal to 5 times the number of hours worked. The amount earned is a function of the number of hours that Amanda worked.
Linear equation y = 5x
Function notation f(x) = 5x
F(1) = 5 ( If we input a value of 1 the output will be 5 because 5(1) =5.
F(3) = 15 ( If we input a value of 3 the output will be 15 because 5(3) = 15.