### Section AEF Trigonometric Functions

**Algebra and Elementary Functions**

**Introduction: **The functions that have been studied in previous sections can be combined with either **
arithmetic operations**, **composition** , or by finding **inverse functions**.
The class of functions that result from using a finite number of
these processes applied to a list of core functions are characterized
as elementary functions. Elementary functions were studied extensively
by Euler and first explicitly defined by Liouville (1837, 1838, 1839).

[See https://www.encyclopediaofmath.org/index.php/Elementary_functions and https://en.wikipedia.org/wiki/Elementary_function ]

**Definition AEF.DEF : Elementary Function Definition**

This section will explore the power of mapping diagrams to make sense of the interactions of the common elementary functions often encountered in precalculus and calculus courses.

AEF.AEFI Elementary Functions are Important. (Not Yet Done)

The following example illustrates with a graph and a mapping
diagram the interaction of core functions in
defining elementary functions through arithmetic operations and
composition.

Example AEF.0**Simple Elementary Function Examples [Graphs and Mapping Diagrams].**

Treatment of elementary functions and their graphical interpretation are familiar subjects for calculus preparation courses.

They often are summarized at the beginning of calculus textbooks and are the major emphasis of the derivative (differential) calculus.

This section presents a more balanced treatment using mapping diagrams to visualize the function operations connecting core functions to elementary functions.

We will develop the basic concepts for elementary functions with mapping diagrams along with comparisons to graphs when appropriate.