Say I have the function $$\tan(\frac{x}{y}) + x\log_{10} (\frac{y}{x}) \cos (\frac{x}{y})$$
I want to know if it is homogeneous. I know it has to satisfy $f(ax,ay)=a^kf(x,y)$, but I don't know if it does.
A short answer/explanation would be appreciated.
It is not homogeneous. Put $f(x, y)=\tan(\frac{x}{y}) + x\log_{10} (\frac{y}{x}) \cos (\frac{x}{y})$. Then for $a \neq 0$ \begin{align} f(ax, ay) &= \tan(\frac{ax}{ay}) + ax\log_{10} (\frac{ay}{ax}) \cos (\frac{ax}{ay}) \\ &= \tan(\frac{x}{y}) + ax\log_{10} (\frac{y}{x}) \cos (\frac{x}{y})\end{align} which is not same to $a^kf(x, y)=a^k\tan(\frac{x}{y}) + a^kx\log_{10} (\frac{y}{x}) \cos (\frac{x}{y})$ in general.
To see this, suppose $f$ is homogeneous and let's find a contradiction. Fix $x=\pi, y=4$ and define $h : \mathbb R \to \mathbb R$ by $$h(a)= \begin{cases}f(ax, ay)=f(a\pi, 4a) & \text{if } a \neq 0 \\ \tan(\frac xy) &\text{if } a=0 \end{cases}$$
Then $h$ is continuous since $$h(0)=\tan \left(\frac xy\right)=\lim_{a \to 0}\left(\tan\left(\frac{x}{y}\right) + ax\log_{10} (\frac{y}{x}) \cos (\frac{x}{y})\right)$$
On the other hand, however, $h(a)=f(ax, ay)=a^kf(x,y)$ for $a \neq 0$ as $f$ is homogeneous. Therefore $$\lim_{a \to 0} h(a)=\lim_{a \to 0} a^kf(x,y)=0$$
Since $\tan(x/y)=\tan(\pi/4)=1 \neq 0 $, we get a contradiction. This proves $f$ is not homogeneous.