Example 

Let f: N → N, under the rule f (x) = a prime factor of x. Is "f" a function?

Solution

Here, As 6 = 2 × 3, f (6) contains two values: f (6) = 2 & f (6) = 3. This rule does not associate with a unique member in the co-domain along a member in the domain and, therefore, f, as defined, is not a function of N into N.

Therefore, you see, to describe a function totally we need to specify the following three things:

(a)        The domain,

(b)        The co-domain, and

(c)        The rule which assigns to each of element x in the domain, a single completely determined element in the co-domain.

Given a function f: X → Y, f (x) ∈ Y is called upon the image of x ∈ X under f or the f-image of x. The set of f-images of all of number of X, for example {f (x): x ∈ X} is said the range of f and is indicated by f (X). It is simple to see that f (X) ⊂ Y.

Remark

(i)         We will consider functions whose domain & co-domain are both subsets of R. Such functions are frequently called real functions or real value functions.

(ii)        The variable x utilized in describing a function is frequently called a dummy variable because it may be replaced by any other letter. Therefore, for instance, the rule f (x) = - x, x ∈ N can as well be written in the form f (t) = - t, t ∈ N, or like f (u) = - u, u ∈ N. The variable x (or t or u) is also called up an independent variable and f (x), that is dependent on this independent variable, is called up a dependent variable.