Domain of an $\mathbb{R}^2$ function involving square root.

Question

Consider the function $f_1(x,y) = \sqrt{\frac{y}{x}}$. The domain is:

$$D_1 = \{(x,y) : (x > 0 \wedge y \geq 0) \vee (x < 0 \wedge y \leq 0)\}.$$

Now, consider the function $$f_2(x,y) = \frac{\sqrt{y}}{\sqrt{x}}.$$

In this case the domain is:

$$D_2 = \{(x,y) : x > 0 \wedge y \geq 0\}.$$

Then I can conclude that $f_1 \neq f_2$??

Answer 1

Yes, as two equal functions must have the same domain.

Answer 2

They are not equal functions, however you can view $f_1$ as an extension of $f_2$, or equivalently, $f_2$ is a restriction of $f_1$, from/to their respective domains.

An extension of a function from a smaller domain to a bigger domain is a function that exists on the bigger domain, and agrees with the old function on all values in the smaller domain. Similarly, a restriction of a function from a bigger domain to a smaller domain is simply the same function but only taking inputs from the smaller domain.