Algebra-5 (CAT-2008)

Let f(x) be a function satisfying f(x)f(y) = f(xy) for all real x,y. If f(2) = 4, then what is the value of f(1/2)?

(1)0   (2)1/4   (3) ½   (4) 1   (5) cannot be determined
Solution follows here:

Solution:

Given that                 f(x)f(y) = f(xy)         f(2) = 4

This problem requires a simple logic that 4 = 2*2 and 2 = 4*(1/2)

As 4 = 2*2, we check f(4):

f(4) = f(2*2) = f(2)* f(2) = 4*4 = 16

As 2 = 4*(1/2),                  

f(2) = f(4 * 1/2) = f(4)* f(1/2) = 16* f(1/2)

=> 4 = 16*f(1/2)     => f(1/2) = 4/16 = 1/4

Answer (2)