According to the Wikipedia article,
[Euler] introduced much of the modern mathematical terminology and notation, particularly for mathematical analysis, such as the notion of a mathematical function.
— Leonhard Euler, Wikipedia
What was the notation for functions before him?
Best Answer
I scanned through parts of Newton's Pricipia found online, and was surprised that a search for the word "function" did not yield any results at all. There do appear to be equations acting as what we would call functions, such as when he describes force, we see things such as $$F=\frac {2h^2}{SP^2}\cdot \frac {QR}{QT^2}$$ and $$R=\frac {\frac 1 2 L}{1+e\cos ASP}$$ (on page 223) but he refers to these as equations, not functions, and admittedly (written the way they are) that is exactly what they are. It seems anything that we would today write as a function, Newton described in words, such as:
Or he used words within his equation:
$$\text{Velocity at P}=\frac {h.VA}{SP^2}$$
This last one almost assuredly would be written as a function if presented in a modern textbook. Newton is certainly not the only source one should consider, but it does give an idea of what was going on right before Euler began publishing.
The information on this website, which unfortunately does not include specific sources, indicates that Bernoulli proposed that $\phi$ or $\phi x$ be used as the notation for a function, and Euler introduced $f(x)$.
Edit: Reference #11 from David Renfro's answer gives references for the statements made about Bernoulli and Euler on the website, as described in the last paragraph above. In my brief skim of Newton's Principia, I also found exactly what was described in reference #11 to be true, specifically that the arguments were motivated almost exclusively from analytic geometry, and that what we consider a "function" was really only considered a variable, as is indicated in the few examples above. I would recommend reading #11, it explains in good detail what you would like to know, I think.