Question 1196484: If  for all a and b, and the function is never equal to zero, find the value of f(5).
Answer by ikleyn(52777)   (Show Source):
You can put this solution on YOUR website! .
If f(2a-b) = f(a)*f(b) for all a and b, and the function is never equal to zero,
find the value of f(5).
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Let solve the problem in 4 (four) steps.
(a) Let a = 1, b = 1. Then, according to the given formula,
f(2*1-1) = f(1)*f(1), or
f(1) = f(1)*f(1).
Since f(1) =/= 0 (as it is given), it implies f(1) = 1 (we can cancel the common factor f(1) in both sides).
(b) Let a = 1, b = 2. Then, according to the given formula,
f(2*1-1) = f(1)*f(2), or
f(1) = f(1)*f(2).
Since f(1) = 1 (we just know it from (a)), it implies f(2) = 1.
(c) Let a = 2, b = 1. Then, according to the given formula,
f(2*2-1) = f(2)*f(1), or
f(3) = 1*1 = 1 (since we just know from (b) and (a) that f(2) = 1, f(1) = 1 ).
(d) Let a = 3, b = 1. Then, according to the given formula,
f(2*3-1) = f(3)*f(1), or
f(5) = 1*1 = 1 (since we just know from (c) and (a) that f(3) = 1, f(1) = 1 ).
Thus we just found out that f(5) = 1, and the solution is complete at this point.
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