I'm publishing my first mathematical paper, and need to create 2D graph figures for representation of a few functions. I've always used Desmos (I used to use Maple for really complex stuff, but I don't have a license anymore), but I don't think that's what you're supposed to use. What program should I use (service preferably free)?
It's worth noting that these functions are quite complex, and when I use a program like Desmos, take 10-20 minutes to render each function.
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Depending on what is needed, I'll screenshot from Desmos and clean it up later. If I need a decent three-dimensional rendering, I may use MATLAB (if you don't have a license, GNU Octave is near-identical), Mathematica, or even GeoGebra. Since your functions are complex, I'd try to use something like Mathematica or MATLAB. Something of this nature greatly depends on what you want to accomplish and what your publishers use. Do you have an editor with whom you can speak about this matter?
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How about Matplotlib? It is the standard graphing tool in my college (along with MATLAB), and can be used online with Google Colaboratory.
For me, its defining feature is the selection of fonts. It ships with a STIX font, which looks like Times New Roman. There's support for a wide range of mathematical symbols (a small subset of TeX). Here's an example of what you can do (with some effort).
This is useful if, for instance, you're writing an IEEE paper: the graph will seem to be a part of the document, being that it uses the same font.
If you don't mind downloading and using it offline (which is easy, because it is a Python package installable via pip), you will be able to use any font whatsoever (so long as the font has glyphs for all mathematical symbols you wish to use).
You can even adjust the DPI setting of the plot. If required, it is possible to save it as an SVG image. For advanced users, Matplotlib can generate code which can be inserted into a LaTeX document to produce a PGF plot.
Most of the commonly used features are extensively documented at https://matplotlib.org/. The only caveat is that you must know Python.
I don't know that there's a particular choice that you're "supposed to make", but I like Sage, which admits programming in Python, has very good graphics stuff, and so on. It's a bit big to download, but I think worth it. There is also an on-line version. See www.sagemath.org