# Definition of Function

A function is a special relationship (correspondence) between two sets such that for each element*x*in its domain there is assigned one and only one element

*y*in its range.

Notice that the correspondence has two parts:

- For each
*x*there is assigned*one**y*. (This is the ordinary part of the definition.) - For each
*x*there is assigned*only one**y*. (This is the special part of the definition.)

The second part of the definition of a function creates the uniqueness of the assignment: There cannot be assigned two values of

*y*to one*x*. In mathematics, uniqueness is very important. We know that 2 + 2 = 4, but it would be confusing if 2 + 2 could also equal something else, say 5. In this case, we could never be sure that the answer to a question was the*right*answer.The correspondence between
For example, the square root function can be written as .
To calculate the correspondence for
For example, the expression becomes the function

*x*and*y*is usually expressed with the function notation:*y*=*f*(*x*), where*y*is called the dependent variable and*x*is called the independent variable. In other words, the value of*y*depends on the value of*x*plugged into the function.*x*= 4, we get . That is, the square root function assigns the unique*y*value of 2 to the*x*value of 4. Most expressions can be turned into functions.