How to find the domain of this function? f(x)= $\frac{(\sqrt{x}-\sqrt{x-1} )}{( \sqrt{x}+\sqrt{x-1} )}\;$
first off $\sqrt{x}$ is defined for:
$$x > 0                   \tag{1}$$
and  $\sqrt{x-1}$ is defined for:
$$x \ge 1                 \tag{2}$$
from $(1)$ and $(2)$, we get domain of $f(x)$ should be $\{x | x\ge 1\}$
We found the domains of elementary functions contained in $f(x)$ and intersected them to find the domain of $f(x)$.
but putting $x = 0$, we get
$$f(0)= -1$$
why this solution was not contained in the above mentioned solution? I always find the domains of composite functions by this way, How can I be sure that some points, like $(0,-1)$ in above problem, are still not included in the solution?

$f$ is a fraction, and fractions are defined everywhere the denominator is non-zero (which it is). The numerator is defined when $[0,\infty) \cap [1,\infty) = [1,\infty)$, (the square root is defined at $0$), and the denominator is defined for
$$
\{x : \sqrt{x} + \sqrt{x-1} \neq 0, \, x\geq 0, x\geq 1\} = [1,\infty)
$$
so the domain is $[1,\infty)$.
By this, we should see that $f(0)$ is not defined.
