In each of the following cases, state whether the function is one-one, onto or bijective. Justify your answer.

Question

In each of the following cases, state whether the function is one-one, onto or bijective. Justify your answer. f : R → R defined by f (x) = 1 + x 2

Philoid · Accepted Answer

It is given that f : R → R defined by f (x) = 1 + x²

Let x₁, x₂ϵ R such that f(x₁) = f(x₂)

Now, f(1) = f(-1) = 2

⇒ f(x₁) = f(x₂) which does means that x₁ = x₂

⇒ f is not one – one

Now consider an element -2 in co- domain R.

We can see that f(x) = 1 + x² is always positive.

⇒ there does not exist any x in domain R such that f(x) = -2

⇒ F is not onto.

Therefore, function f is neither one-one nor onto.