Question

Is the domain of F(x)= 2 + 3^x all real numbers.....or all real numbers >0?

Answer 1

The domain of 3^x is real and greater than zero for all real values of x.
Therefore dom F(x)=2+3^x is all real and greater than zero.
See graph of F(x) attached. 

Answer 2

Focus on y=3^x first.  The domain of this function includes all real numbers, positive, zero or negative.  When complex numbers are considered, the domain includes complex numbers in additionl to all real numbers.

Thus, the domain of F(x)= 2 + 3^x consists of all real numbers.  Certainly F(x) is defined for negative x values.