Rewrite your function in terms of x and y. Then solve for y in terms of x:
y = -1/2 * (sqrt(x+3)
-2y = sqrt(x+3)
4y^2 = x +3
x = 4y^2-3
Now swap x and y.
y = 4x^2-3
So you got the correct answer. To determine the domain, look back at the original function:
y = -1/2 * (sqrt(x+3), x ≥ -3
We see that x must be greater or equal to -3 in order for the square root to be a real number. Thus sqrt(x+3) must be nonnegative, by that I mean positive or zero.
So if that part of our function is positive and then we are multiplying it by -1/2, our function will only output a nonpositive number, by that I mean either negative or zero. That means our range for this function is y ≤ 0.
And by definition of the inverse, the range of the original function becomes the domain of the inverse. That means the domain of our inverse is x ≤ 0.
I hope that makes sense!
