$f(x) \neq x$ ambiguity? The question is :
An integer function $f(x)$ is valid only for $x = 0, 1, 2, 3$ and has an interesting property $f(f(x)) = x$. It is also known that $f(x) \ne x$. Find out how many such functions exist.
This problem was on a mock mathcounts test that we did as a team at MATHCOUNTS Practice. My solution is as follows:
We have two cases:
Case 1: 0 goes to 1, 1 goes to 0, 2 goes to 3, 3 goes to 2. There are $\dbinom42 / 2 = 3$ ways this configuration can occur (we simply must form two groups of two)
Case 2: 0 goes to 1, 1 goes to 0, 2 goes to 2, 3 goes to 3. There are $\dbinom42 = 6$ ways; we simply choose the two that are paired.
We don't need to consider 0 - 0, 1 - 1, 2 - 2, 3 - 3 because $f(x) \ne x$.
However, the answer was $3$ and the given solution was simply "$\dfrac{\dbinom42}{2} = 3$". Not very explanatory, but from this we can see that they intended a different meaning of $f(x) \ne x$ then I interpreted.
I took $f(x) \ne x$ meaning the function, $f(x)$, is not equal to the function $f(x) = x$. In my mind, two polynomials are equal if their difference forms the zero polynomial. For example, $f(x) = x$ and $f(x) = 2x + 1$ are not equal, though they are equal at $x = -1$.
When I brought up this point in the practice, my coach dismissed it, claiming the question was "not ambiguous at all".
This is why I am making this post; my question is not whether my answer $9$ to the given question is correct. It is merely whether the question is ambiguous in any sort of way. According to me, the question should have either specified their intended definition of $\ne$, or given an example of an acceptable configuration that would clear up the doubt. However, my coach believes that the question contains no ambiguity what so ever, the question is clear as it is, and $3$ is the one and only correct answer.
So what answer would you give to this question? why? and most importantly, is the question, in any way, ambiguous?

Usually a predicate of the kind $P(x)$, when no quantifier is explicitly provided on $x$, should be interpreted as $\forall x\, P(x)$. When you have an equality, like $f(f(x))=x$ this is clear. I agree with you that $f(x)\neq x$ is a little bit less clear... but it should be interpreted as $\forall x\, f(x)\neq x$. I would have no doubt about it.
To write that two functions are different, one would write $f\neq g$ instead of $f(x)\neq g(x)$, because the function is $f$ not $f(x)$ (which is the value of $f$ at some point $x$). In your case one should have written $f\neq id$ to mean $\exists x\, f(x)\neq x$ as you have interpreted.
