What is the difference between
- functional relationship
- functioning relationship
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Merriam-Webster list these three "simple definitions" of functional...
1: designed to have a practical use
2: (medical) affecting the way a part of your body works
3: working properly
...where sense #3 is synonymous with functioning, but as implied by @Damkerng T.'s comment above obviously it's possible something that's expected to be of practical use (be functional) might not actually be working properly. So it's possible to have a non-functioning functional relationship.
I would just add that I think M-W's first definition above is slightly more restrictive than it should be. A relationship might simply arise (or be identified) that was primarily noted for its functional effects even if it wasn't actually "designed" (by engineers, God, evolution, etc.). And at any given point in time that relationship might (temporarily) not be performing as expected (not necessarily as designed).
Going further in that direction, I could also imagine using functional relationship to describe a relationship that was based on functionality. For example, relationships with others in the workplace might arose solely because of the functions you and they perform at work (and in some cases those relationships might not even have any practical use, for either the workers or the employer).
That's to say, functional can have a range of meanings (broadly, having to do with functionality), whereas functioning only ever means working [properly].
By a quick dictionary search you can find out that the word "functional" is an adjective. In fact, there are many words with -al endings that are adjectives, such as electrical, divisional, internal... There is not much more to add to that, it's just a typical every-day adjective.
Now, the word "functioning" is an adjective as well. It is actually something that is called present participle adjective. Verbs have the present participle form, and you may remember it from continuous tenses. It is formed by adding the ending -ing to verbs e.g. functioning, working, swimming, praying... These kinds of adjectives have a special use:
"Present participles (-ing) are used to describe the people or things that cause the feelings."
Quoted from here.
You can find similar explanations on other sites as well, such as this one.
So to take some examples, "boring news", "annoying noise", "terrifying movie" would respectively mean news that cause boredom, noise that causes annoyance, movie that causes terror. Consequently, "functioning relationship" would mean a relationship that causes function, which doesn't really make sense to me. So you should always say "functional" relationship, not "functioning".