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