solve-the-equation-of-function-f-3x-2-f-2x-2-f-x-2-

Question Number 112681 by bemath last updated on 09/Sep/20

solve the equation of function

{(f (3 x))}^{2} = {(f (2 x))}^{2} + {(f (x))}^{2}

Answered by bobhans last updated on 09/Sep/20

(⧫) {(f (3 x))}^{2} = {(f (2 x))}^{2} + {(f (x))}^{2}

by letting h (x) = {(f (x))}^{2}

now we get h (3 x) = h (2 x) + h (x)

plugging in x = 0

\Rightarrow h (0) = 2 h (0), \Rightarrow h (0) = 0

since the equation is linear, let h (x) = Cx

therefore f (x) = \pm \sqrt{Cx} or f (x) = A \sqrt{x}

where A can be any constant

Facebook Tweet Pin