how can we prove that a function is onto? please explain that if range = codomain then the function is said to be onto. how?

Consider a function

The set of all elements of X are called the domain of f and the set of all elements of Y are called the range of f.

Range is also called as image of f.

 

In general, the image of f is a subset of codomain.

 

It may coincide with its codomain.

 

A function f from X to Y is called onto if for all y in Y, there is an x in X, such that

So to prove that a function is onto, we need to prove that the range of the function is equal to its codomain.

 

Now let us consider the function,

This function is a straight line.

 

 

Suppose

If x=1, then we have, f(x) = x + 1 = 2

If x=2, then we have, f(x) = x + 1 = 3

.

.

.

.

 

Thus, each x value has one unique y=f(x) value.

 

That is for all elements in the range is having their pre-images in domain.

 

Thus we have, range = codomain