Say I've got a function $x_1(p_1,p_2,m)$ where $p_1, p_2$ are the prices for good 1, good 2 respectively and m is the income.
Now, I haven't heard of the Slutsky equation yet nor the income/substitution effect.
How do I show for this function that it is an inferior/normal and ordinary/Giffen good?
My initial thought was to simply use the partial derivate and to check whether it's greater or less than zero, since
$\frac{\partial x_1(...)}{\partial m} \gt 0$ should be true for normal goods, $\frac{\partial x_1(...)}{\partial m} \lt 0$ for inferior goods, $\frac{\partial x_1(...)}{\partial p_1} \gt 0$ for Giffen goods and $\frac{\partial x_1(...)}{\partial p_1} \lt 0$ for ordinary goods.
Is this correct?
Is this all I can do to formally show that the good is normal/ordinary? Is this sufficient?
Also, if I can show that the same function for $x_1(m)$ is homogeneous of degree 1 (i.e. homothetic), i.e. $tx_1(m)=x_1(tm)$, this means that the partial derivative $\frac{\partial x_1(...)}{\partial m}$ is homogeneous of degree 0, so it would be a constant. Could I thus conclude that the good is strictly normal/inferior (is that a term?), i.e. it doesn't change if the income changes?