Proving that a function is not invert-able (one way function) I am having problems with proving if the one-way function (https://crypto.stackexchange.com/tags/one-way-function/info) is hard to invert or not.

$2^\sqrt {m} $ one-way function  $ f: \{0, 1\}^{2m} \to \{0,1\}^{2m} $ 

and $ g(x) = f(0...0^m || x_1, x_2, \ldots, x_m) $  $ g: \{0, 1\}^{m} \to \{0,1\}^{2m} $ (the || means here concatenation). Is function g one-way and what could be told about its security?
I think you can prove $g$ is not guaranteed to be one-way by the following counter example. First let $S$ be the set of all strings in $\{0,1\}^{2m}$ that starts with $m$ zeroes. Take any OWF $f$ of the type you describe. Consider the function $f'$ that is identical to $f$ except on all inputs $x \in S$ it acts as the identity function. I.e. for all $x \in S$ we have $f'(x) = x$. Now if you use $f'$ to construct $g$ then $g$ will clearly not be one-way (because this means $g$ is just the identity function). So all you need to prove is that $f'$ is still a OWF.

Now I sketch an argument that $f'$ is in fact a OWF. Since $f'$ only changes the output of $f$ on a negligible amount of inputs, the distribution of output of $f'$ and $f$ on a uniformly random input should be statistically indistinguishable. However, if there exists an adversary $A$ that demonstrates that $f'$ is not a OWF, then that adversary would work as a distinguisher between the two output distributions. Thus,  $~f'$ must be a OWF.

I am not sure this is 100% watertight, so you may want to work through the proof yourself :).