For what values of x is √x+3 defined?

√x + 3 is a pair of real numbers for all values of 'x' that are zero or greater.

√(x + 3) is a pair of real numbers for all values of 'x' that are -3 or greater.

For other values of 'x', there is a whole system of so-called imaginary numbers
that you will learn about in future math classes.  With those,  you'll be able to
handle square roots of things that may sometimes be negative.