Is$\frac{\sqrt{a}}{\sqrt{b}}$ the same as $\sqrt{\frac{a}{b}}$?

Question

My idea is that the two functions are not the same since for the first function, the domain of the function is only non negative reals for the numerator and positive reals for the denominator. However in the second function, the domain is any real provided $b$ isn't equal to $0$.

Am I correct in thinking so?

If they are indeed the same function, please do explain why their domains will be the same?

Answer 1

Let us define the functions: $$ f(a,b)=\frac{\sqrt{a}}{\sqrt{b}}\quad\,\,\text{and}\,\,\quad g(a,b)=\sqrt{\frac{a}{b}}. $$ Then $f$ and $g$ AGREE on the intersection of their domains. However, they have different domains:

$$ \mathrm{Dom}(f)=\{(a,b): a\ge 0,\,\,b>0\}, $$ while $$ \mathrm{Dom}(g)=\{(a,b): a\ge 0,\,\,b>0\}\cup\{(a,b): a\le 0,\,\,b<0\}. $$

Strictly speaking, in order for two functions to be equal they need to have the same domain (and they same values for each element of their domain.) Hence, strictly speaking, these functions are not the equal.

Answer 2

Actually, you are right- as functions of two real variables, they have different domains. For example take $a=-1,b=-1$

Answer 3

The question demonstrates the difference between expressions and functions. And the question of finding the maximal domain for a function determined of an expression depends on what one mean with the expressions in the formula. Either is $\sqrt x$ a function defined for $x\geq 0$ or it is an other function, defined for other domains. Anything goes but has to be declared.