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AP EAMCET202119 Aug 2021Evening ShiftMathematicsFunctionsActual

Let f : R → R be a function defined by f ( x ) = 4 x 4 x + 2 . what is the value of f 1 4 + 2 f 1 2 + f 3 4 = ?

Options

  1. A4
  2. B3
  3. C2
  4. D1

Correct answer

C. 2

Step-by-step solution

Given, f ( x ) = 4 x 4 x + 2 We will show that for the given function f ( 1 - x ) = 1 - f ( x ) f ( 1 - x ) = 4 1 - x 4 1 - x + 2 = 4 · 4 - x 4 · 4 - x + 2 = 4 4 + 2 · 4 x = 4 x + 2 - 4 x 2 + 4 x = 4 x + 2 2 + 4 x - 4 x 2 + 4 x = 1 - 4 x 2 + 4 x = 1 - f ( x ) ⇒ f 1 4 = f 1 - 3 4 = 1 - f 3 4 Now, f 1 4 + 2 f 1 2 + f 3 4 = 1 - f 3 4 + 2 f 1 2 + f 3 4 = 1 + 2 f 1 2 = 1 + 2 4 1 2 4 1 2 + 2 = 1 + 2 × 2 4 = 2

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