The inverse of a function f is a function f^(-1) such that f^(-1)(f(x)) = x for every x in the domain of f.
One trick for finding the inverse of a function f is to set x=f(y) and then solve for y. The resulting equation will be of the form y=f^(-1)(x).
If we apply that trick to the given function, though, we run into an issue:
Let x = h(y) = 4-y^2
=> y^2 = 4-x
=> y = +-sqrt(4-x)
It seems, then, that we would have h^(-1)(x) = +-sqrt(4-x). Unfortunately, this is not a function, as by definition, a function can only produce one value for any given input. Here we rely on the additional information given in the problem.
We are given that x<=0, meaning that h^(-1)(h(x)) <= 0. This tells us that the inverse function should be producing negative values, meaning we want -sqrt(4-x), rather than sqrt(4-x). Thus, our final answer is
h^(-1)(x) = -sqrt(4-x)
(As a side note, we do not have to worry about taking the square root of a negative number as the domain of h^(-1) is the same as the range of h, which is (-oo, 4]. Thus we will not apply h^(-1) to any value greater than 4.)
