Definition and terminology of a function

What is a function for?

In science, economics or physics, we constantly encounter quantities that depend on one another: the distance travelled depends on time, the price of a taxi journey depends on the number of kilometres, and the area of a square depends on its side length. A function is the mathematical tool that describes this relationship: for every input value, it assigns a single output value.

Basic vocabulary

• The input variable is called the antecedent, often denoted by x.
• The associated value is called the image, denoted by f(x) (read as ‘f of x’).
• The function is denoted as f: x → f(x).

Example: if f(x) = 2x + 1, then the image of 3 is f(3) = 2 × 3 + 1 = 7. We say that 3 has 7 as its image, and that 7 has 3 as its antecedent (under f).

x	-2	0	3
f(x) = 2x + 1	-3	1	7

Common pitfall

Do not confuse f and f(x): f is the name of the function (the ‘machine’), whilst f(x) is the number obtained when x is fed into this machine. Writing “f = 2x + 1” is incorrect; you must write “f(x) = 2x + 1”.

Uniqueness of the image

A given function can only produce one image for a given argument. Conversely, the same number may well be the image of several different arguments: for example, f(x) = x² gives f(2) = 4 and f(-2) = 4. Two arguments, a single common image: this is permitted.