The domain of composite functions?

Question

The following is a composite function: g of f

f(x) = sqrt(2x+3)
g(x) = x^2+1

(sqrt(2x+3))^2+1

As it is, I think the domain is [-3/2, Infinity)

The weird thing is, I could reduce this to 2x+4 where the domain would instead be all real numbers.

Could someone confirm for me that it is nescessary to first: find the domain before reducing. And also, is it proper to reduce composite functions since it changes the domain?

ironduke, sqrt(neg. number) = error

Answer 1

f(x) = sqrt(2x+3)
g(x) = x^2+1
g(f(x)) = (sqrt(2x+3)^2+1) = 2x+4

In either case the domain is all the real nunbers.