f : R→ R defined by f(x) = 1 + x^2

Question

Check the injectivity and surjectivity of the following functions: 

f : R→ R defined by f(x) = 1 + x²

Answer 1

Let x₁, x₂ ∈ R 

f(x₁) = f(x₂) 

⇒ 1 + x₁² = 1 + x₂² 

⇒ x₁ = ±x₂ 

∴ f is not one-one 

Also range of f contains only real numbers which is greater than or equal to 1≠ R 

∴ f is not onto hence f is not bijective.