Show that f: R → R defined as f(x) = x / sqrt(1 + x^2) is one-one but not onto.
Show that f: R → R defined as f(x) = x / √(1 + x²) is one-one but not onto. f: R → R, f(x) = x / √(1 + x²) To check One-One / Injective Let x₁, x₂ ∈ R such that f(x₁) = f(x₂) x₁ / √(1 + x₁²) = x₂ / √(1 + x₂²) --------[1] Squaring both sides: x₁² / (1 + x₁²) = x₂² / (1 + x₂²) x₁²(1 + x₂²) = x₂²(1 + x₁²) x₁² + x₁²x₂² = x₂² + x₁²x₂² x₁² = x₂² x₁ = ± x₂ [1] is possible only if x₁, x₂ have the same sign x₁ = -x₂, is rejected we have to conclude that x₁ =x₂, Therefore f is one-one. To check Onto / Surjective Let y = x / √(1 + x²) squaring y² = x² / (1 + x²) y²(1 + x²) = x² y² + y²x² = x² y² = x² - y²x² = x²(1 - y²) x² = y² / (1 - y²) For x to be real, RHS ≥ 0 Since y² ≥ 0, we need 1 - y² > 0 ⇒ y² < 1 ⇒ -1 < y < 1 Range of f = (-1, 1) ≠ R Therefore f is not onto. cbse 12th maths old board exam question paper 2025 2026 one to one injective function onto function
