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The domain of the function f(x) = ⁻¹ ( 3 ⁻¹ ( 2x 1+x^2 ) ) is

Options

  1. A[- 3 , - 1 3 ] [ 1 3 , 3 ]
  2. B(- , - 3 ] [- 1 3 , 1 3 ] [ 3 , )
  3. C(- , 1 3 ] [ 3 , )
  4. D[- 1 3 , 1 3 ]

Correct answer

B. (- , - 3 ] [- 1 3 , 1 3 ] [ 3 , )

Step-by-step solution

For the function f(x) to be defined, the argument of the outer ⁻¹ function must lie in [-1, 1] . -1 3 ⁻¹ ( 2x 1+x^2 ) 1 - 3 ⁻¹ ( 2x 1+x^2 ) 3 Applying the sine function, which is strictly increasing in this interval: - 3 2 2x 1+x^2 3 2 Since 1+x^2 > 0 for all real x , we can multiply throughout without changing the inequality signs: - 3 (1+x^2) 4x 3 (1+x^2) This gives a system of two inequalities: Inequality 1: 4x 3 x^2 + 3 3 x^2 - 4x + 3 0 ( 3 x - 1)(x - 3 ) 0 x (- , 1 3 ] [ 3 , ) Inequality 2: - 3 x^2 - 3 4x 3 x^

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