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.