So, a function tells you how to match up one group of numbers with another group of numbers. For example, suppose 2 cans of soup cost $3.00. Then, 3 cans would cost $4.50, and 4 cans would cost $6.00. See how the numbers match up?
2 cans matches up with $3.00
3 cans matches up with $4.50
4 cans matches up with $6.00
Or, if you want to forget about what the numbers mean, then...
2 matches with 3.0
3 matches with 4.5
4 matches with 6.0
That rule for matching one certain number of cans with another certain number of dollars can be expressed in at least three ways.
(1) As a data table:
Number of  Cost of Soup Cans  Soup + 1  $1.50 2  3.00 3  4.50 4  6.00 5  7.50
(2) As a graph:
8.00 +  X +  6.00 + X  + Cost of  X Soup 4.00 +  ($) + X  2.00 +  X +  0.00 X++++++ 0 2 4 6 Number of Soup Cans
(3) Or, as an equation, here where the variable c stands for the cost of the soup, and the variable n stands for the number of soup cans:
c = 1.5n
INPUT and OUTPUT
Consider this simple function:
y = 2x
The variable x is where a number comes into the function. Therefore, we could call x the input variable.
The function rule says to multiply the number in x by 2 and then put this result, or output, into the variable y.
The variable y, therefore, could be called the output variable.
DOMAIN and RANGE
Again, consider this simple function:
y = 2x
The group, or set, of numbers that "goes into" a function is called the DOMAIN of the function.
The set of numbers that "comes out of" a function is called the RANGE of the function.
All of the various numbers that could possibly go into x make up the group of numbers called the domain of the function. In this example, since x could accept any real number, we would say that the domain of this function is all real numbers.
All of the various resultant numbers that end up in y make up the group of numbers called the range of the function. In this example there is exactly one real number that ends up in y for every one real number that went into x. So, we would say that the range of this function is also all real numbers.
Since x could be set equal to 3, we would say that the number 3 is in the domain of the function. The number 3 in x would be multiplied by 2, making a 6. This 6 would go into y. Since the 6 is output by the function, we would say that 6 is in the range of the function.
DOMAIN = INPUT NUMBERS = x
RANGE = OUTPUT NUMBERS = y
We have been looking at this function:
y = 2x
Notice that it has an equal sign in it, so it is an equation, or it could be called a relation.
However, not everything with an equal sign in it is a function.
Technically speaking, not every equation is a function. We need to find out when an equation is a function and when it is not.
The above type of function has a special name.
It is called a ONETOONE FUNCTION.
The above type of function also has a special name.
It is called a MANYTOONE FUNCTION.

The above type of equation
has a special name, too.
It is called a ONETOMANY RELATION.
Be sure to understand that a onetomany relation is not a function.
Below are several videos that will show you how ONETOONE FUNCTIONS, MANYTOONE FUNCTIONS, and ONETOMANY RELATIONS appear when they are seen as graphs. In each one the xaxis is shown in blue, the yaxis in red, the function (or relation) in yellow, and points and construction lines in white.