How would you find the largest possible domain in real numbers for function

Questions
1

such as this function
(x^3 + 5 x^2 + 4 x + 5)/Sqrt[x^2 + 2 x - 3]

2

Resolve[Exists[y, y == (x^3 + 5 x^2 + 4 x + 5)/Sqrt[x^2 + 2 x - 3]], Reals]

Out[10]= x < -3 || x > 1
3

What is that input essentially saying?

4

I want to know for what values of x there will exist real y such that y=f[x], where f is your function. Per documentation on Resolve, it does the needed quantifier elimination.

5

In V10 there is also

In[1]:= FunctionDomain[(x^3 + 5 x^2 + 4 x + 5)/Sqrt[x^2 + 2 x - 3], x]

Out[1]= x < -3 || x > 1