No Problemmo #2

Algebra Level 5

Let $f(x,y)$ be a periodic function satisfying $f(x,y) = f( 2x+2y , 2y-2x )$ for all $x,y$ belonging to the set of real numbers .

Let $g(x) = f(2^x , 0)$ . Then find the fundamental period of $g(x)$ .

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8 6 12 None of the others Insufficient data 48 24 2

1 solution

Sibasish Mishra
Jan 20, 2017

f(x,y) = (2x+2y, 2y-2x)
= (2(2x+2y)+2(2y-2x), 2(2y-2x)-2(2x+2y))
=(8y, -8x)
=(8(-8x), -8(8y))
=(-2^6x, -2^6y)
=(-2^6(-2^6x), -2^6(-2^6y))
=(2^12x, 2^12y) =f(x,y)
=(2^24x, 2^24y) = f(x,y)
Now g(x) = f (2^x,0) = (2^12(2^x), 0)
=(2^(x+12),0)
= (2^(x+24),0) = g(x+12)
Hence, g(x) = g(x+12)
So, the period of g(x) is 12.