Q. 1 B5.0( 4 Votes )

# Give an example of a functionWhich is not one – one but onto.

TIP: One – One Function: – A function is said to be a one – one functions or an injection if different elements of A have different images in B.

So, is One – One function

a≠b

f(a)≠f(b) for all f(a) = f(b)

a = b for all Onto Function: – A function is said to be a onto function or surjection if every element of A i.e, if f(A) = B or range of f is the co – domain of f.

So, is Surjection iff for each , there exists such that f(a) = b

Now, Let, given by f(x) = x3 – x

Check for Injectivity:

Let x,y be elements belongs to R i.e such that

So, from definition

f(x) = f(y)

x3 – x = y3 – y

x3 – y3 – (x – y) = 0

(x – y)(x2 + xy + y2 – 1) = 0

As x2 + xy + y2 ≥ 0

therefore x2 + xy + y2 – 1≥ – 1

x – y≠0

x ≠ y for some Hence f is not One – One function

Check for Surjectivity:

Let y be element belongs to R i.e be arbitrary, then

f(x) = y

x3 – x = y

x3 – x – y = 0

Now, we know that for 3 degree equation has a real root

So, let be that root  Thus for clearly , there exist such that f(x) = y

Therefore f is onto

Hence, given by f(x) = x3 – x is not One – One but onto

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