Consider the following C code segment

int f (int x)
{
      if (x < 1)  return 1;
      else return (f(x-1) + g(x))
}

int g (int x)
{
      if (x < 2) return 2;
      else return (f(x-1) + g(x/2));
}

Of the following, which best describes the growth of f(x) as a function of x?

(A) Linear
(B) Exponential
(C) Quadratic
(D) Cubic


Answer: (B)

Explanation: f(n) = f(n-1) + g(n) ---- 1
g(n) = f(n-1) + g(n/2) ---- 2

Putting the value of g(n) in equation 1,
f(n) = 2*f(n-1) + g(n) + g(n/2)

So, we can derive the below equation,
f(n) > 2f(n-1)
=> f(n) > 2*2*f(n-2) ---- because f(n) > 2*f(n-1), so, f(n-1) > 2*2*f(n-2).... so on
=> f(n) > (2^n)f(1) --- here \'^\' denotes the exponent.
So, f(n) > Theta(2^n)

So, option B is true, exponential growth for f(x).

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