I am trying to understand cases that require using functors. Most of the answer on Stackoverflow and other websites put emphasis on being able to define different adders or multipliers regarding benefits of functors.
Can the use of functors go beyond them? What are some other uses of functors?
More often than not, functors are used with other API calls that need some kind of function object. For example, sorting vectors of user-defined objects which don't have operator() or operator< (etc.) defined.
There are some cases where a set of functors may prove useful. One such case comes when you have several algorithms which functionally do the same thing, but achieve varying levels of accuracy. This happens a lot with some numeric optimization problems: given the general form of a matrix, we might use a different technique to find the solution of a linear equation (e.g., sparse vs dense problem-matracies can employ different algorithms to invert the matrix).
In particular, you should consider functors versus lambdas. In modern versions of C++, there really isn't a need to specify a functor unless you're implementing a function/method that needs a functor (or lambda) as an argument. There are some cases to consider: Do you need a unit-test? Is the functor itself a prototype of future functionality? etc.
ADDENDUM: The key thing to consider is that the use of functor/lambda ultimately boils down to a design decision. As @t.niese noted in the comments, you could use just use functions in combination of template arguments. In addition to the previous considerations above, consider whether or not you can make a compile-time or run-time assessment of the needed functionality.
Additionally, as you make design decisions, you may want to consider "Is there a need for this function to be used outside of this specific context?" If the answer is no, that's a compelling argument to choose a lambda over a free function. With regards to functor specifically, this was an important pattern added before the addition of lambdas to the standard. Typically they're defined in a somewhat private context (frequently in the implementation files, thus after compiled into a library, obfuscated to users of the API). Now with lambdas, you can simply define them within another function or even as a function argument, instead of pre-defining them prior to need.
