Here is a somewhat different approach to this topic than Han Sah's. When I read the original post asking about "elementary functions," I thought perhaps it was from a parent whose child was taking a course in high school with such a name (there are quite a few). So, I was initially reminded of an old resource: the 1959 Report of the Commission on Mathematics entitled "Program for college preparatory mathematics." [Yes, I still have a copy on my shelf :)]
Back then, one recommendation for a one semester course in the 12th grade was an "Elementary Functions" course containing the following topics: I. Sets and combinations 1. Review and extension of concepts 2. Permutations and combinations 3. Mathematical induction II. Functions and relations 1. Sets of ordered pairs 2. Functions 3. Relations 4. Inverse relations and functions III.Polynomial functions 1. Review of linear and quadratic 2. The general polynomial 3. Slope of a graph at a point 4. Slope function 5. Polynomial equations IV. Exponential Functions 1. Review of definition, properties, and graph over the rationals 2. Extension to the reals 3. Exponential growth V. Logarithmic functions 1. Review of definition, properties, and graph of 10^x and inverse 2. Extension to base a>0 [but not 1] 3. Graphs VI. Circular functions 1. Radian measure 2. Definition of sine x and cos x for real numbers 3. Graphs 4. Inverses 5. Solution of trigonometric equations 6. Power series
I have ommitted outline details in the interest of time. Also, the Commission also published an Appendix for Teachers which contains information, instruction, and enrichment ideas that help in translating the recommendations of the Report into action. The contents also amplify and clarify some of the recommendations. A useful 225-page resource.
At present, I note that some precalculus texts, such as the North Carolina "Contemporary Precalculus" [Janson] and some calculus texts, such as the Harvard Consortium materials [Wiley] make extensive use of a "Toolkit of Functions" or a "Library of Functions," respectively. These include: constant, linear, quadratic, cubic, square root, absolute value, reciprocal, and sine functions in the Toolkit initially, with polynomial, rational, exponential, logarithmic, and circular functions coming later. Also, transformations, compositions, and inverse functions are treated as well following the Toolkit, for example.
So, when some people think of "elementary functions," I think they are referring either to a senior-level high school course similar to the one described above, or to the collections of functions necessary for the study of calculus.
Ron Ward/Western Washington U/Bellingham, WA 98225 firstname.lastname@example.org