One-one Function

Let the function h: x → x2, mentions on the set R. Here h (2) = h (- 2) = 4; for example 2 and - 2 are distinct members of the domain R, but their h-images are the similar (Can you discover some more members whose h-images are equivalent?). Generally, it may be expressed by saying: discover x, y such that x ≠ y but h (x) = h (y).

Now let another function g : x → 2x + 3. Here you shall be able to see that if x₁ & x₂ are two different real numbers, then g (x₁) and g (x₂) are also different.   

For,

x₁ ≠ x₂  ⇒ 2x₁ ≠ 2x₂ ⇒ 2x₁ + 3 ≠ 2x₂  + 3 ⇒ g ( x₁ ) ≠ g ( x₂ ) .

 Here we have considered two functions. When one of them, namely g, sends different members of the domain to different members of the co-domain, the other, namely h, does not always do this. We give a special name to functions as g above.

Definition

A function f: X → Y is stated to be a one-one function (a 1-1 function or an injective function) if images of distinct members of X are distinct members of Y. Therefore, the above function g is one-one, while h is not one-one.

Remark

The condition that the images of different members of X are different members of Y in the above definition may be replaced by either of the following equivalent conditions:

(a)   For each pair of members x, y of X, x ≠ y ⇒ f (x) ≠ f (y).

(b)   For each pair of members x, y of X, f (x) = f (y) ⇒ x = y. We have earlier observed that for a function f: X → Y, f (X) ⊆ Y.

This opens two possibilities which are following:

(a)        f (X) = Y, or

(b)        f (X) ⊂ Y, that is f (X) is a appropriate subset of Y.

The function h: x → x₂, for all of x ∈ R falls in the second category. As the square of any real number is always non-negative, h (R) = R+ ∪ (0) that is the set of non-negative real numbers. Therefore, h (R) ⊂ R.

Alternatively, the function g : x → 2x + 3 belongs to the first category.

Given any y ∈ R (co-domain), if we assume x = (1/2) y - 3/2, we discovers that

g (x) = y. This indicates that every member of the co-domain is a g-image of some member of the domain and, therefore, it is in the range g. From this, we obtains that g (R) = R. The following definition characterises this property of the function.