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.
If you write $f : A \to B$ that is usually understood to mean that:
- for every $a \in A$, the value $f(a)$ is defined
- this value is an element of $B$.
So the domain has to be $A$. However, $B$ is not necessarily the image but just the codomain (i.e. it is acceptable for $f$ to not “hit” every value in $B$; however, if $f$ does "hit" every value in $B$, then the image is equal to the codomain).
For your definition $f(x) = \sqrt{1 - x^2}$, writing $f : \mathbb{R} \to \mathbb{R}$ would be considered wrong in most circumstances (people get sloppy sometimes, or figuring out the exact domain might be something they want you to do; writing $f : \mathbb{R} \to \mathbb{R}$ still sloppy in the latter case, though). So you have to be more precise in your domain, i.e. $f: [-1, 1] \to \mathbb{R}$ would be fine (note: you do not have to specify the image, just a superset of it), as would $f : [0,1] \to [0, 1]$ would (i.e. you can consider the function restricted to a subset of its maximal domain).
Best Answer
One often sees $\operatorname{dom }f$ and $\operatorname{ran} f$ or $D(f)$ and $R(f)$ or $D_{f}$ and $R_{f}$ .
Of course, you can also say something like
or whatever. You're allowed to make up whatever notation you like, as long as you explain it clearly.