### SectionAEFTrigonometric 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).

The class of elementary functions includes the rational functions, the rational power functions, exponential and logarithmic functions, and the trigonometric and inverse trigonometric functions.
[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.