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Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution.
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The first step of the VA (Variable Approach) method is to modify the original condition and the question. We then recheck the question. We should simplify conditions if necessary.
Since we have many variables to determine the function f(x), E is most likely the answer. So, we should consider conditions 1) & 2) together first. After comparing the number of variables and the number of equations, we can save time by considering conditions 1) & 2) together first.
Conditions 1) & 2)
Since f(3) = 5, we have:
f(5) = (f(3) - 1) / (f(3) + 1)
f(5) = (5 - 1) / (5 + 1)
f(5) = 4/6 = 2/3.
Then we have:
f(7) = (f(5) - 1) / (f(5) + 1)
f(7) = ((2/3) - 1) / ((2/3) + 1)
f(7) = -(1/3) / (5/3) = -(1/5).
We have:
f(9) = (f(7) - 1) / (f(7) + 1)
f(9) = (-(1/5) - 1)/(-(1/5) + 1)
f(9) = -(6/5) / (4/5) = -(3/2).
We have:
f(11) = (f(9) - 1) / (f(9) + 1)
f(11) = (-(3/2) - 1)) / (-(3/2) + 1)
f(11) = -(5/2) / -(1/2) = 5.
Since f(3) = f(11), we have f(8k - 5) = 5.
Then we have:
f(8k-3) = 2/3, f(8k-1) = -(1/5) and f(8k+1) = -(3/2).
f(2007) = f(8*251-1) = -(1/5).
Since both conditions together yield a unique solution, they are sufficient.
Therefore, C is the answer.
Answer: C
