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

Asked by sreenanda dhar | 1st Jan, 2014, 12:24: PM

Expert Answer:

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
 
 
 
 
 
 

Answered by Vimala Ramamurthy | 2nd Jan, 2014, 09:54: AM

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