What would be to teach in most countries and education systems?
(Taught before "high education frames", i.e. before doing bachelor of arts in mathematics).
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The linear function is a pretty good first example: they are very fundamental and everything is simple. But rather than think in terms of the most fundamental thing to come next, I would encourage you to think in terms of typical examples that show the breadth of the concept.
For signal processing you might go more in the direction of sine and cosine as simple examples and in general use more periodic examples.
The category-theorist in me would argue the most fundamental function is the identity function, so $f(x)=x$ might be a good one to go to next.
I say this one because, when you really start diving down into what "fundamental" is, the existence of an identity mapping is really key to a lot of structures. In particular, categories. And, of course, its a dirt simple function, which is probably what you're going for.
Of course this being said, the constant function and identity function can also be tricky first functions because they are so simple that it's hard to get any feel for functions at all with them.