Input–output behaviour of functions

Many functions can be described by some kind of expression, e.g. x2 + 1, that tells us how, given an input value for x, one can calculate an output value. In the present case this proceeds as follows: first determine the square of the input value and consequently add 1 to this. The so-called ‘variable’ x acts as an arbitrary (or abstract) input value. In a concrete case, for example when using input value 3, one must replace x with 3 in the expression. Function x2 + 1 then delivers the output value 32 + 1, which adds up to 10.

In order to emphasise the ‘abstract’ role of such a variable x in an expression for a function, it is customary to use the special symbol λ: one adds λx in front of the expression, followed by a dot as a separation marker. Hence, instead of x2 + 1, one writes λx · x2 + 1, which means ‘the function mapping x to x2 + 1’. This notation expresses that x itself is not a concrete input value, but an abstraction. As soon as a concrete input value comes in sight, e.g. 3, we may give this as an argument to the function, thus making a start with the calculation.