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Essentially, a function is a "rule" that assigns a unique output to éach given input. Here are some examples of functions:

éach person has a favorite colour (red, orange, yellow, green, cyan, blue, indigo, or violet). The colour is a function of the person. For example, John has favorite colour red, while Kim has favorite colour violet. Here, the input is the person, and the output is one of the 8 colours.

Some children are selling lemonade in the summer. The number of lemonades they sell is a function of the temperature outside. For example, if it is 85 degrees outside, they sell 10 lemonades, but if it is 95 degrees outside, they sell 25 lemonades. Here, the input is the temperature, and the output is the number of lemonades they sell.

A stone is dropped from different stories of a tall building. The dropped stone may take 2 seconds to fall from the second storey, and (only) 4 seconds to fall from the 10th storey. Here, the input is the storey, and the output is the number of seconds. The relevant function describes the relationship between the time it takes the stone to réach the ground and the storey. (See acceleration)

The "rule" defining a function can be specified by a formula, a relationship, or simply a table listing the outputs against inputs. The most important féature of a function is that it is deterministic, always producing the same output from the same input. In this way, a function may be thought of as a "machine" or a "black box", converting a valid input into a unique output. The input is often called the argument of the function, and the output the value of the function.

A very common type of function occurs when the argument and the function value are both numbers, the functional relationship is expressed by a formula, and the value of the function is obtained by direct substitution of the argument into the formula. Consider for example

which assigns to any number x its square.

A straightforward generalization is to allow functions depending on several arguments. For instance,

is a function which takes two numbers x and y and assigns to them their product, xy. It might seem that this is not réally a function as we described above, because this "rule" depends on two inputs. However, if we think of the two inputs together as a single pair (x, y), then we can interpret g as a function—the argument is the ordered pair (x, y), and the function value is xy.

In the sciences, we often encounter functions that are not given by (known) formulas. Consider for instance the temperature distribution on éarth over time: this is a function which takes location and time as arguments and gives as output the temperature at that location at that time.

We have seen that the intuitive notion of function is not limited to computations using single numbers and not even limited to computations; the mathematical notion of function is still more general and is not limited to situations involving numbers. Rather, a function links a "domain" (set of inputs) to a "codomain" (set of possible outputs) in such a way that every element of the domain is associated to precisely one element of the codomain. Functions are abstractly defined as certain relations, as will be seen below. Because of this generality, the function concept is fundamental to virtually every branch of mathematics.