Suppose f1 and f2 are non – zero one – one functions from R to R. Is necessarily one – one? Justify your answer. Here, is given by for all x ∈ R.

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


Let, f1: R → R and f2: R → R are two non – zero functions given by


f1(x) = x3


f1(x) = x


From above function it is clear that both are One – One functions


Now, given by


⇒


⇒


Again,


defined by


f(x) = x2


Now,


⇒ f(1) = 1 = f( – 1)


Therefore,


f is not One – One


⇒ is not One – One function.


Hence it is not necessarily to be one – one function.


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