Let A = [–1, 1]. Then, discuss whether the following functions defined on A are one-one, onto or bijective:
k(x) = x²

Question

Let A = [–1, 1]. Then, discuss whether the following functions defined on A are one-one, onto or bijective: k(x) = x 2

Philoid · Accepted Answer

(iv) Given that, A = [–1, 1]

let k(x₁) = k(x₂)

⇒ x₁²= x₂²

⇒ x₁= ± x₂

⇒ x₁= x₂ and x₁= - x₂

For e.g., k(-1) = |-1| = 1 and k(1) = |1| = 1

⇒ k is not one-one.

We observe that -1 does not have any pre-image in the domain since k(x) = x² assumes only non-negative values.

i.e. we cannot find any number in domain which will give -1 in co-domain.

⇒ k is not onto

⇒ Hence, k is neither one one nor onto.

Let A = [–1, 1]. Then, discuss whether the following functions defined on A are one-one, onto or bijective:k(x) = x2