Q. 5 I5.0( 2 Votes )

# Classify the foll

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

Bijection Function: – A function is said to be a bijection function if it is one – one as well as onto function.

Now, Let, given by f(x) = x3 + 1

Check for Injectivity:

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

So, from definition

f(x) = f(y)

x3 + 1 = y3 + 1

x3 = y3

x = y

Hence f is One – One function

Check for Surjectivity:

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

f(x) = y

x3 + 1 = y

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

Thus, It is also Bijective function

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