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AP EAMCET202119 Aug 2021Evening ShiftMathematicsFunctionsActual

The real valued function f ( x ) = x e x - 1 + x 2 + 1 defined on R 0 is

Options

  1. AAn odd function
  2. BAn even function
  3. CBoth even & odd function
  4. DNeither even nor odd function

Correct answer

B. An even function

Step-by-step solution

Given f ( x ) = x e x - 1 + x 2 + 1 = 2 x + x e x - x + 2 e x - 2 2 ( e x - 1 ) = e x ( x + 2 ) + x - 2 2 ( e x - 1 ) Then, f ( - x ) = e - x ( - x + 2 ) - x - 2 2 ( e - x - 1 ) = - x + 2 e x - x - 2 2 1 e x - 1 = - x + 2 - x e x - 2 e x 2 1 - e x = - x + 2 - e x x + 2 2 1 - e x = x - 2 + e x x + 2 2 e x - 1 ⇒ f ( x ) = f ( - x ) Therefore, f ( x ) is an even function.

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