What was the notation for functions before Euler? 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?
 A: 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:

If a hyperbolic orbit be described under the action of a repulsive
  force tending from the center, the force varies as the distance and
  the velocity at any point as the diameter of the conjugate hyperbola
  parallel to the tangent at the point.

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.
A: The question is ill-posed, as there was no concept of function before Bernoulli and Euler in anything approaching the modern sense. 
Leibniz and Bernoulli mostly worked with variable quantities.  Thus, a curve may be defined by an equation involving both $x$ and $y$ and the functional relation between $x$ and $y$ was only implicit, because neither was chosen as the independent variable upon which the other would depend.  Similarly, what we would today call differentiation involved relations among infinitesimal differentials $dx$ and  $dy$.  What we would today express as the idea of independent variable was expressed by saying that $dx$ varies in constant increments.
Since there was no concept there could not have been a notation for it. Thus, in the 17th century mathematicians mainly worked with curves defined by an equation, and studied there properties. This does not require anything close to the modern notion of a function and they did not have one, though Leibniz did realize the functional relationship between $x$ and $y$ along a curve.
A: Let's observe an example :
$a)$ formal description of function (two-part notation)
$f : \mathbf{N} \rightarrow \mathbf{R}$
$n \mapsto \sqrt{n}$
$b)$ Euler's notation :
$f(n)=\sqrt{n}$
I don't know who introduced two-part notation but I think that this notation must be older than Euler's notation since it gives more information about function and therefore two-part-notation is closer to correct definition of the function than Euler's notation.
There is also good wikipedia article about notation for differentiation.
