The function f: R R defined by f(x) = x x^2+1 x R is

Step-by-step solution

Given f(x) = x x^2+1 To check for one-one, let f(x₁) = f(x₂) x₁ x₁^2+1 = x₂ x₂^2+1 x₁ x₂^2 + x₁ = x₂ x₁^2 + x₂ x₁ x₂(x₂ - x₁) - (x₂ - x₁) = 0 (x₂ - x₁)(x₁ x₂ - 1) = 0 This gives x₁ = x₂ or x₁ x₂ = 1 . For example, f(2) = 2 5 and f ( 1 2 ) = 2 5 . Since f(2) = f ( 1 2 ) for 2 1 2 , the function is not one-one. To check for onto, let y = x x^2+1 y x^2 - x + y = 0 For x to be real, the discriminant must be non-negative: 1 - 4y^2 0 y^2 1 4 - 1 2 y 1 2 The range of f(x) is [- 1 2 , 1 2 ] , which is a proper subset of th