Question

Show that the function f : R → R : f (x) = x² is neither one-one nor onto.

Answer 1

To prove: function is neither one-one nor onto

Given: f : R → R : f (x) = x²

Solution: We have,

f(x) = x²

For, f(x₁) = f(x₂)

⇒ x₁² = x₂²

⇒ x₁ = x₂ or, x₁ = -x₂

Since x₁ doesn’t has unique image

∴ f(x) is not one-one

f(x) = x²

Let f(x) = y such that \(y\in R\)

⇒ y = x²

⇒ \(x=\sqrt{y}\)

If y = -1, as \(y\in R\)

Then x will be undefined as we cannot place the negative value under the square root

Hence f(x) is not onto

Hence Proved