Best way to determine domain/codomain of a function?

Given a function what is the best way to determine domain/codomain and or image of a function?

Is there any particular methods on how best to approach it or is it just looking at the sets (which has been mainly real to real or natural to natural) and seeing which parts need certain conditions (can't divide by zero or root a negative) and then seeing what possible outputs you can get for the function to determine codomain?

Also is there anything else for more complicated functions is it worth splitting them up into composite functions?

For the domain typically take all numbers, and exclude ones that won't work in the function. For the codomain, this is usually just the real numbers but someone could correct me on this.

On another note-does anyone else find it weird how students are being asked to specify the domain and codomain? I always thought it was the sort of thing you are given because functions don't make sense without them being predefined.

My impression (and I'm willing to be corrected on this) is that since the single A level content has been reduced from the level of the pre-modular A level, this has led to a subtle change of emphasis with a disproportionate focus on certain topics in a (possibly misguided) attempt to identify the "better" candidates through the use of what I would call "awkward questions" rather than "mathematically useful" questions.

So, for example, instead of building up candidates' confidence with tricky differentiations or integration by substitutions, we invent the concept of domain and range being separated from the definition of function, and then students find themselves confronted with mysterious textbook questions like this "Explain why 1/(x-2) isn't a function". Then three lessons further on, the book will say "Differentiate the function 1/(x-2)"!

A similar sort of thing seems to be happening with graph transformations. Now, everyone should understand basic horizontal and vertical translations of graphs, and it's useful to understand the periodicity and symmetry properties of the trig graphs. But we now see increasingly elaborate questions asked about identifying sequences of transformations in functions like ln(4 - 3x), where the function has no "natural scale" or "natural periodicity" and it's not entirely clear why, when or how you would every apply a transformation to it!

Not sure I've answered your question, but just felt like a minor rant