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Let f(x) = 9^x 9^x + 3 and let g(x) be the inverse function of f(x) . Then the value of _ k=1 ¹²⁰ g ( k 121 ) is equal to

Options

  1. A120
  2. B60
  3. C61
  4. D0

Correct answer

B. 60

Step-by-step solution

Given f(x) = 9^x 9^x + 3 Evaluate f(1-x) : f(1-x) = 9^ 1-x 9^ 1-x + 3 = 9 9^x 9 9^x + 3 = 9 9 + 3 9^x = 3 3 + 9^x Adding f(x) and f(1-x) : f(x) + f(1-x) = 9^x 9^x + 3 + 3 9^x + 3 = 1 Let y = f(x) . Since g(x) is the inverse function of f(x) , we have x = g(y) . From f(1-x) = 1 - f(x) = 1 - y , we get 1 - x = g(1-y) . Adding these two relations gives: g(y) + g(1-y) = x + (1 - x) = 1 This means g(x) + g(1-x) = 1 for all x in the domain of g . The required sum is S = _ k=1 ¹²⁰ g ( k 121 ) S = g ( 1 121 ) + g ( 2 121 )

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