I lurk in a lot of math and education forums.
It’s amazing how ‘precalculus’ seems to have such a wide variety of topics, depending on the school.
I’ve always assumed it was a class aimed at college students who stopped at Algebra II in high school. Mostly a review of more difficult algebra topics as well as learning or reviewing trigonometry.
But, it seems it could be defined as very basic algebra all the way up to taking derivatives.
So, what do you consider to be precalculus?
Generally, quadratic equation, inequalities, some trig.
The prerequisites for the first calculus course I took were “college algebra” and “elementary functions.”
College algebra covered things like rational functions, logarithms, polynomials, and systems of equations (among other things). Elementary functions focused on trig. All of these things were utilized in calculus so I would expect a pre-calculus course to cover at least these topics.
Perhaps an introduction to limits, the definition of the derivative, and some basic polynomial differentiation could also be included, but that would make the first 2-3 weeks of calculus something of a refresher.
In addition to the aforementioned topics, pre-calc may cover complex numbers and their relationship to trig (roots of unity, etc.)
I’m tutoring a high school student in pre-calc now. It seems to go into more detail in the analytical geometry than I remember when I took pre-calc. That may be because I took it in a college class which covered both trig and pre-calc in one semester, in contrast to this year-long high school class.
To me, “precalculus” means the analysis, including drawing the graph, of real functions of a single real variable but without explicitely using tools from differential calculus. So essentially polynomial functions (especially the quadratic, but the general asymptotic behaviour of other polynomial functions should be taught as well), some rational functions (especially finding their roots and asymptotes), simple algebraic functions such as the square root, and the common transcendental functions such as the exponential, logarithmic, and trigonometric, and possibly the inverse trigonometric as well. The idea is that the student would gain experience analysing functions of a single variable before being taught a more general method using derivatives in a subsequent calculus class.
As someone who’s taught calculus courses, this is what I expected (or at least hoped) my students had seen before entering my classroom. Maybe other subjects are also taught in courses labelled “precalculus”, but to me the above are the important ones. Of course, the student should also know about factoring polynomials, doing operations on fractions, etc., but those are probably seen before precalculus.
I’d expect a lot of analytic geometry, introducing the relationship between functions and the shapes that arise when you graph them.
This is almost the exact setup where I teach - even the course names. But we don’t do systems of equations in College Algebra (they come earlier in the sequence), and there is an emphasis on being able to draw graphs by hand. Besides the basic trig stuff, Elementary Functions includes introductions to polar coordinates and vectors.
I speak as one who has taught a class called “Precalculus” at the college level. At my institution (which I think is fairly typical), Precalculus is taken after College Algebra and before Calculus I.
The majority of the course is trigonometry. (At my previous institution it was in fact Trigonometry that was the course offered between College Algebra and Calculus I). I can break that down further if you’d like.
But it also includes a review of topics involved in functions in general, and exponential and logarithmic functions. (These topics are covered in College Algebra but we review them and take them a bit further in Precalculus.) And it includes an introduction to the geometry of complex numbers, polar coordinates, and conic sections.
I believe this set of topics is fairly typical for a course called “Precalculus” at the high school or lower college level, but the exact details may certainly vary.
When I was in high school myself, in the early 80s, I took a class called Precalculus that was basically half a year of College Algebra and half a year of Trigonometry.
I think “precalculus” suggests “immediately before calculus”, and not merely “before calculus” which would for example include learning to count.
So, I’d expect limits would figure heavily. Likewise perhaps the concept of slopes in graphs and the use of small differentials or limits in estimating slope.
Introducing derivatives and for example differentiating polynomials certainly isn’t precalculus. It’s calculus, if only the beginning.
Agree with your first and third paragraphs. But limits would also be part of calculus itself, rather than pre-calculus.
I agree that limits are squarely in the realm of the infinitesimal calculus. In fact, I would expect a rigorous first calculus class (as opposed to “calculus for engineers” 
to go over and not skip the construction of the real numbers, sequences and convergence, limits, infinite sums, real functions and continuity, differentiation and Riemann integration, etc. These are all real, live calculus and not merely “precalculus”.
There are a number of textbooks titled “Precalculus”; e.g. the first one I clicked on covers “functions” (domain and range, graphs of functions, composition, inverse functions, etc.), linear functions, polynomial functions, rational functions, exponential/logarithmic/trigonometric functions, systems of equations and inequalities, analytic geometry, probability, and properties of limits, continuity and derivatives. So it seems like more or less a collection of topics that college students are expected to be sort of familiar with when beginning to study “calculus”, even if technically speaking there are no prerequisites beyond “elementary” mathematics. For instance, how to define the exponential function of real numbers and its properties of continuity and smoothness is definitely calculus, but the students are ideally supposed to be familiar with taking powers of at least integers and rational numbers.
Not surprisingly, the Summary for that book explicitly states:
More evidence, if any is needed, that not all classes called “Precalculus” will cover the exact same set of topics.
And yes, it is possible (but doesn’t always happen) that a Precalculus class would include at least an introduction or sneak peak at topics that are actually part of calculus.
Your use of “rigorous” reminds me of the preamble to a calculus text I have from the 1960s (Analytic Geometry and the Calculus):
It seems to me that a really rigorous book on the Calculus would begin with the axioms of set theory, derive the Peano axioms, and then reproduce most of Landau’s great Foundations of Analysis. After this preparation, the class (if there are any students left) would have no trouble following the proofs needed in the Calculus.
As humorous as the preface is, this is still the same sort of textbook that recommends (without blinking) in one footnote that the student reads a 200-page text on conic sections, and includes several instances of the statement “the proof thereof is trivial and left as an exercise for the reader.”
See, all that stuff is obviously Precalculus
I remember that my high school precalc class was explicitly split between trigonometry and analytical geometry.
I’m not groking these references to college pre-calculus. I may even have “college” wrong.
Is “college” the senior years of high school?
No. It’s just that some classes (including pre-calculus) are offered in high school, but some students do not take them in high school and have to take them in college.
It could be, but the U.S. also has many “community colleges” as well as the traditional collegiate divisions of state and private universities. Not sure it’s important in the context of this discussion; “college” algebra, “college” geometry, “college” pre-calculus and so on are, as far as I can tell, referring to prerequisite modules that first-year university students are supposed to have studied, whether in high school or at a separate educational institution.
Everything in a precalc class could logically fit into some other class, whether algebra, trig, or calculus itself. There are a set number of topics to fit in before you take calculus, and a set number of classes to cover them in. Most schools will cover the same topics, in the same number of classes. The question then just becomes, which topics to group together, and what to call the resulting classes. Both of which are, comparatively speaking, relatively trivial questions.
Ah. So “college” is university, but “college math” is math you study at high school if you’re in a college stream.
Where I am, when I was young, “maths” was college stream, and “business maths” was terminal.