AM-GM Inequality Basic question

Since $xy>0$ $\displaystyle{\frac{x}{y}}>0$ $\displaystyle{\frac{y}{x}}>0$

So we can use the Arithmetic Mean-Geometric Mean Inequality:

$\frac{\frac{x}{y}+\frac{y}{x}}{2}\geq\sqrt{\frac{x}{y}\times\frac{y}{x}}=1$

$\therefore f(x)\geq2$