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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:
  1. For each x there is assigned one y. (This is the ordinary part of the definition.)
  2. 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 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.
For example, the square root function can be written as .
To calculate the correspondence for 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.
For example, the expression  becomes the function

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