Construct a computable function from non-computable funciton
There are many ways to do this, here is one example.

To make $h$ uncomputable, you can choose $f_1$ to be the constant $1$ function, and $f_2$ to be any non computable binary function, i.e. whose range is $\{0,1\}$. In that case $h=f_2$ is not computable. In fact, when $f_1,f_2$ are binary functions, $h$ is never computable (why?).

To make $h$ computable you can let $f_1$ be the constant $1$ function, and $f_2$ be any non computable function whose range is $\mathbb{N}\setminus\{1\}$. In that case $h$ is zero everywhere.