Mathematics - Functions Question with Solution | TestHub

$f(x,y)=f(2x+2y,2y-2x)=f(2(2x+2y)+2(2y-2x),2(2y-2x)-2(2x+2y))$

$$\begin{array}{rl}& =f(8y,-8x)\\ & =f(8(-8x),-8(8y))=f(-64x,-64y)\\ & =f(-64(-64x),-64(-64y))\\ & =f\left(2^{12}x,2^{12}y\right)\\ & f(x,y)=f\left(2^{12}x,2^{12}y\right)\\ & f(x,0)>f\left(2^{12}x,0\right)\\ & g(x)=f\left(2^x,0\right)=f\left(2^{12}{2}^x,0\right)\\ & =f\left(2^{x+12},0\right)=g(x+12)\\ & \Rightarrow g(x)=g(x+12)\\ & \Rightarrow \text{ period of }g(x)\text{ is }12=T\\ & \Rightarrow \frac{T}{4}=3\end{array}$$