Computability of function composition I have some problems to understand computability and hope you can help me.
In the lecture we had following problem:
Consider the three partial functions $f,g,h\colon N \to N$, where $f$ is computable and $g$ is not computable. The following two statements are correct:

If $h(x) = g(f(x))$, then $h$ could be computable.

If $g(x) = h(f(x))$, then $h$ is not computable.


Well, 2 is clear I think, because we know that $g$ is not computable and $f$ is computable, so $h$ has to be not computable, otherwise $g$ would be computable.
But 1 is a big problem for me. If $f$ is computable and $g$ is not computable, how can $h$ be computable?
Suppose that $f(x) = 0$. Then $g(f(x))$ is computable for any function $g$, computable or not. This explains 1.
As for 2, the composition of two computable functions is computable, so if both $f$ and $h$ are computable, so is $g$. Since $f$ is computable and $g$ isn't, the only conclusion is that $h$ is not computable.