It's only meant as an example, so I'll take a different example... if we let $f(x)$ be the unary representation of $x$, where $x$ is a number given in binary, then $f(x)$ always exists (so the question whether it exists is trivial), but is time-intensive to compute.

In $Mx=0$, assuming $x\ne 0$, there are two separate questions: is there such an $x$, and how do you find one if there is one. Conceivably the former could be easy while the latter is hard, so maybe some clarification of the question?

@Andrej Bauer: Yes, (1) is a good restatement of the inequality above. (2) is also true but only because of (1).

en.wikipedia.org/wiki/Chebyshev_polynomials