How would you Approach the Function?

Algebra Level 3

If, for all real x x , f ( x ) = 2 x f ( 1 − x ) f(x)=2^xf(1-x) , what is the numerical value of f ( 3 ) f(3) ?


The answer is 0.

Danish Ahmed by Danish Ahmed , 24, India

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2 solutions

Aareyan Manzoor
Dec 31, 2015

put x with 1-x to get f ( 1 − x ) = 2 1 − x f ( x ) f(1-x)=2^{1-x}f(x) 2 x f ( 1 − x ) = f ( x ) = 2 x − 1 f ( 1 − x ) 2^xf(1-x)=f(x)=2^{x-1}f(1-x) from this we get 2 f ( 1 − x ) = f ( 1 − x ) 2f(1-x)=f(1-x) or f ( 1 − x ) = 0 f(1-x)=0 and f ( x ) = 0 f(x)=0 at the case of interest: f ( 3 ) = 0 f(3)=\boxed{0}

2 Helpful 0 Interesting 0 Brilliant 0 Confused
Danish Ahmed
Dec 31, 2015

Plugging in 3 3 and − 2 -2 into this equation we find that

f ( 3 ) = 8 f ( − 2 ) f(3)=8f(-2) and f ( − 2 ) = f ( 3 ) 4 f(-2)=\dfrac{f(3)}{4}

Thus we have f ( 3 ) = 2 f ( 3 ) ⟹ f ( 3 ) = 0 f(3)=2f(3)\implies f(3)=\boxed{0}

0 Helpful 0 Interesting 0 Brilliant 0 Confused

how do you know a function f(x) exists satisfying the question? you must show that or your solution will be incomplete. @Danish Ahmed

Aareyan Manzoor - 5 years, 5 months ago

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