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

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

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

Answer»

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