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COMEDK202510 May 2025Morning ShiftMathematicsInverse Trigonometric FunctionsActual

The range of x for which the equation ⁻¹ ( 2 x 1+x^2 )=2 ⁻¹ x holds true

Options

  1. Ax 0
  2. Bx R
  3. C|x| 1
  4. D|x| 1

Correct answer

D. |x| 1

Step-by-step solution

The expression ⁻¹ ( 2x 1+x^2 ) is defined for | 2x 1+x^2 | 1 , which implies 2|x| 1+x^2 , or (|x|-1)^2 0 , which is true for all x R . Recall the identity ⁻¹ ( 2x 1+x^2 ) = 2 ⁻¹ x for |x| 1 . For x > 1 , let x = where ( /4, /2) . Then 2x 1+x^2 = (2 ) . Since 2 ( /2, ) , ⁻¹( (2 )) = - 2 = - 2 ⁻¹ x . Thus, the identity does not hold for x > 1 . For x Therefore, the equation holds true if and only if |x| 1 . Answer: |x| 1

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