It's even worse than you describe it!
f needs to be linear, but the function in your example is not linear.
However, there are quite interesting linear functions. Example: f(x(t)) = x(t-2) + 4dx/dt - \int_0^t 2x(s) ds
f needs to be linear, but the function in your example is not linear.
However, there are quite interesting linear functions. Example: f(x(t)) = x(t-2) + 4dx/dt - \int_0^t 2x(s) ds
5(2z) + 2 != 2(5z + 2)
For example, linear functions over finite dimensional vector spaces can be represented with matrices which means that everything you can compute about matrices you can also compute about linear functions.