-たい words and a student's confusion

Question

I was doing some Rosetta Stone earlier, and I came upon a sentence ending with:

行きたいです。

Noting the "たい" part, I was reminded of the word

食べたいです

which I think I first saw in Japanese 1010, and I'm sure I've read and said while doing Rosetta Stone. At this point, my supposition was that "...行きたいです is to 行きます as 食べたいです is to 食べます."

Observing the "です" ending, I suspected that (1) 行きたい is a noun, which jisho.org confirmed...albeit only if I searched for "いきたい." If I search for "行きたい," it pares my search down to "行く." And (2) I suspect that these "たい" forms might be a common occurrence, as a way to make the "I want to [do some verb]" form of a word.

I was then a bit surprised to see that "食べたい" (or "たべたい") wasn't in the dictionary. Google Translate recognized it and translated it to "want to eat," so I imagine I wrote it correctly, and all that.

So then, my questions:

1. 食べたい is a word, right? Rosetta Stone hasn't been feeding me made-up words? I'm keenly aware that the reason it doesn't come up in jisho.org might just stem from jisho.org's search engine, but I figure this question is still worth asking. (Of course, since Google Translate translated it, I'd be surprised if the answer is anything but 'yes, of course it's a word.')
2. Was I wrong to infer that these "たい" forms (forgive me if they have a better name, or if I'm completely mis-classifying them) are built on the same roots as the verbs? I have no encountered many of them yet...and a sample size of "two words" isn't a lot to go on, I realize. Still, it's in my nature to start trying to infer things, as early in a learning process as I can manage.

3. Is it possible that 食べたい is some type of conjugation (of yet another word) that I simply haven't heard of yet? "たい" does come up in jisho.org as an 'auxiliary adjective,' which this entry:

   1: (after the -masu stem of a verb) want to ... do something; would like to ...; (Particle)

   This suggests that the answer to my 2nd question is that, yes, there's a clearly-defined process for taking the root of a verb, putting -たい at the end of it, and calling it the 'want to [verb]' form.

Honestly, after putting this question together, I'm starting to think that the answer might look like this: "(1) Yes, it's a word; it just doesn't have an explicit entry in jisho.org. (2) No, that inference is ok, as you surmised after looking up "たい" and seeing that first entry. (3) Yes; as you said, it uses the -たい ending. Really, it's just time for you to learn more about these 'auxiliary adjectives.'"

I'd prefer to hear it from someone more knowledgeable then myself, of course. ^-^

One more question, to tack on: (4) Are these -たい words always nouns?

Answer 1

1. yeah it's a word. 食べたい and 食べます are considered forms of 食べる, which is the only one that will have an explicit entry in a dictionary.

2. you're fine. the たい forms are built from the "masu" stem of the verb (Vmasu) by taking off the ます and adding たい.

3. my answer to (1) should answer this too. the たい words are conjugations of verbs such as 食べる. the definition you found for たい is appropriate for this discussion.

4. these たい words aren't nouns. they conjugate the same way as い-adjectives like いい and ふるい. for this reason, i believe they're considered い-adjectives, although i mentally place them in a slightly different category. the reason you thought these words were nouns is because you saw them followed by です. but です can actually follow い-adjectives.

ない is also listed as an auxiliary adjective by jisho.org and it turns verbs into い-adjectives just like たい does (although it uses a different verb stem).