Question

A function f:R⁺ → (0, 1) is defined as f(x) = 1/(x² + 1). Prove that f is a bijective function.

Answer 1

Given f(x) = 1/(x² + 1)

⇒ f'(x) = 2x/(x² + 1)² > 0, ∀ x ∈ R⁺

⇒ f is strictly increasing function.

⇒ f is one-one function.

Also, the range of a function is (0, 1)

⇒ R_f = (0, 1) = co-domain

⇒ f is onto function

Thus, f is a bijective function.