Was the Forbidden Forest a part of Hogwarts?

YES

In Harry Potter and the Sorcerer's Stone we have this quote from Dumbledore, "the forest in the grounds is forbidden to all pupils." In is the key word there. A castles "grounds" are the outdoor area around the castle that is the property of the owner. Think a houses' front or backyard. So it's pretty clear that the Forbidden Forest is a part of Hogwarts.

As to your follow-up question "what's the purpose" that's never blatantly stated. However it seems reasonable to assume it's kept as a sort of "magical plant/animals preserve" for Hogwarts, much like a Medieval castle would have a forest on the grounds where the nobility could hunt. At least three magical creatures, Unicorns, thestrals, and Bowtruckles, appear to both live in the forest AND be subject to study by students at Hogwarts. (Unicorns are mentioned numerous times in the books as living in the forest and studied in Harry Potter and the Goblet of Fire, and in Book 5 Harry Potter and the Order of the Phoenix they study thestrals that are kept in the forest and pull the school carriages. Also in book 5 (iirc) a bowtruckle used in Care of Magical Creatures escapes back to the forest.) It may be that these and other creatures are too rare/elusive/temperamental to be taken from some other location and brought to Hogwarts for study by students, so instead they're kept in the Forbidden Forest for easy access.

Another potential reason for the forest being "in the grounds" of Hogwarts is that Hogwarts itself is a powerfully magical location. Muggles can't see it for what it is, and it's home to a bunch of enchantments and other protective spells. It may be safer to keep a forest in this location for the use of magical creatures like centaurs than risk centaurs running about "in the wild" and being spotted. It also may be easier to set up such a preserve in the relatively large enchanted grounds of Hogwarts than bother setting up a similar site somewhere else in Britain for Centaurs et al to use as a haven. This is more speculation than anything based on hard facts though.

Dumbledore repeatedly says the forest "on the grounds" is off-limits:

"As ever, I would like to remind you all that the forest on the grounds is out-of-bounds to students, as is the village of Hogsmeade to all below third year." (GOF, "The Triwizard Tournament")"

"First years should note that the forest on the grounds is forbidden to all pupils. And a few of our older students would do well to remember that as well." (PS, "The Sorting Hat")

This has two potential meanings. Is the forest on the grounds off-limits, or the forest the the grounds off-limits? One would suggest that part of the forest rests on Hogwarts grounds, and the other suggests the entirety of the forest is on the grounds. But either way it seems clear that part of the forest lies on Hogwarts grounds.

Complicating matters is that there's a colony of centaurs residing in the forest who seem convinced it is their forest, and every time they say so to a human, the humans respond, nah.

'As are you, human,' said Bane, 'coming back into our Forest when we warned you--'

'Now, yeh listen ter me,' said Hagrid angrily. 'I'll have less of the "our" Forest, if it's all the same ter yeh. It's not up ter yeh who comes an' goes in here--'

'No more is it up to you, Hagrid,' said Magorian smoothly. 'I shall let you pass today because you are accompanied by your young--' (OOTP, "Grawp")

'What are you doing in our Forest?' bellowed the hard-faced grey centaur Harry and Hermione had seen on their last trip into the Forest. 'Why are you here?'

'Your Forest?' said Umbridge, shaking now not only with fright but also, it seemed, with indignation. 'I would remind you that you live here only because the Ministry of Magic permits you certain areas of land--' (OOTP, "Fight and Flight")

Again, this can all be interpreted different ways. Is this Ministry land that centaurs are permitted to live on? Land that the Ministry permits the centaurs to own? Hogwarts-owned land that has been set aside by Ministry for centaur use? If it's not up to Hagrid-- the groundskeeper-- to say who enters the forest, does that mean it's not on the grounds, or that it's up to Dumbledore or the Ministry? Or is Magorian's retort just a complete denial of human claims entirely?

But in both interactions, the text says that Harry and co. travelled very deep into the forest. I lean towards the explanation that at some point the part of the forest that is on Hogwarts grounds ends, and Ministry and/or centaur owned land begins. But this is admittedly a guess.

So I'm going with the text is ambiguous and also the question itself is politically fraught, similar to asking about, say, the status of Jerusalem or the Western Sahara.

I'm going to add to what other have said about the forest being part of the castle grounds, and explain why I think the forest ended up filled with magical creatures, and whether or not it was planned.

Like with all animals, some magical creatures prefer living in forests. Presumably creatures like Bowtruckles can be found all over the world, in places where there are trees of wand quality. I'm not sure about Thestrals, but I assume they too could thrive in other forests. Them living in the Forbidden Forest could therefore just be accidental. There would be less risks posed by Muggles, like deforestation, hunting and picnicking, but otherwise it would be very much a normal forest.

That said, Bowtruckles (and most likely Thestrals too, despite being considered dangerous) doesn't pose as much of a threat to Muggles as other creatures. Dragons for instance (classified as XXXXX), while there as none of those living in the forest as far as I know, are kept in close control by the Ministry. Dragons are only permitted in certain areas (though of course they don't care about that themselves, they fly wherever they want), usually far from Muggles (mentioned either in GoF or Fantastic Beasts I think). Most likely, dragons would be considered too dangerous for them to live in the forbidden forest.

It is possible that similar protected areas exist for other creatures, both in the cases where the creatures are dangerous and in cases where they need protection. The Forbidden Forest could be one such area where endangered animals are kept. One such species is unicorns, which are both dangerous and possibly endangered. The grounds at Hogwarts are magically protected, which would make it harder (but not impossible) for poachers to hunt down magical animals, like the unicorns, to collect potion ingredients without regard to the creature's survival.

Another creature classified as dangerous (XXXX) by the Ministry is centaurs. They have near-human intelligence, at least according to the Ministry, and are very protective of their territory. They would therefore need to live somewhere far from Muggles, or Muggles could end up being hurt. Being intelligent and long lived creatures, they would likely be far more picky with where to live than Thestrals, and would not like being told what to do by humans.

We do not know exactly how the centaurs ended up in the forest. One may assume that, being a magical place, some centaurs might have decided on settling there centuries ago. Later, the Ministry probably wanted the centaurs to live in a controlled area, while the centaurs would only move voluntarily. They could possibly then later have come to agreement, so that more/most/all centaurs moved to Hogwarts. While less than ideal to have dangerous creatures close to the school, it was probably decided to be the better solution, since it would be easier to keep a watch on them there.

To sum up, I believe a good explanation is that there wasn't an intention for the forest to be filled with magical creatures, it just happened over the course of centuries. Some animals probably lived there before the school opened, some have moved there since either of their own accord or was moved by force by the Ministry. Most of them are either mostly harmless, or dangerous only when angered, which is probably the reason students are supposed to keep out of the woods.

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Another interesting observation I've noticed, is that seemingly all creatures I have checked so far that live in the forest are classified as XXXX (with one exception), which means dangerous but not impossible to tame or handle if you are experienced. The exception being the Acromantulas, Aragog's family. If I recall, Aragog was placed there (very deep in the forest) by Hagrid on suggestion from Dumbledore, which I assume wouldn't be popular with parents if they knew. Acrumantulas are considered untameable, and while Hagrid (and probably Dumbledore, being as trusting as he was) assumed that the spiders would keep to themselves, we clearly see on several occasions (CoS and HBP) that while Aragog cares about Hagrid, the spiders in general have no problem attacking/eating students or other humans from the school.