The elementary functions are defined from an initial list of core
functions.These core functions can be characterized as being
either i) algebraic or ii)transcendental.**
**

**
(i) Algebraic Core Functions.
(ii) Transcendental Core Functions.
**

**(i) Algebraic Core Functions. **

The core algebraic
functions can all be related to the constant function, $f_k(x) = k$ where
$k$ is a real number, and the identity function, $id(x)= x$. From these
two functions using the basic arithmetic of functions a second level of
core functions are polynomial and rational functions. Using inverses to
enlarge the algebraic core functions one can also include functions that
are rational powers, $f(x) = x ^{p/q}$ where $p/q$ is a rational
number. These functions have been discussed in previous sections, especially, Section 4 Other
Algebraic Functions (OAF).

For many purposes of instruction, a simple list for core algebraic
functions is given as $f \in \{f_k,id,x^n\}$ where $k$ is a real number,
and $n$ is any integer.

**(ii)Transcendental Core Functions.**

The core transcendental elementary functions can all be related (using
complex numbers) to the exponential function, $\exp(x)= e^x$.
What is important about these functions is that they can not be
expressed as finite combinations of the algebraic functions. [This
result is not easy to prove. See Wikipedia:Transcendental_function]

Using the basic arithmetic with complex functions, one can develop a
second level of transcendental core
functions: the circular and hyperbolic trigonometric functions of sine,
cosine, hyperbolic
sine, and hyperbolic cosine. Using the basic arithmetic of real values
functions, the next level of core functions are the remaining
trigonometric and hyperbolic trigonometric functions, Finally by using
inverses and some simple compositions to enlarge the transcendental core
functions one can include all core exponential and logarithmic
functions and the inverse trigonometric and inverse hyperbolic
trigonometric functions. These functions have been discussed in previous
sections, especially, Section 6
Exponential and Logarithmic Functions (ELF) and Section 7 Trigonometric Functions (TRIG)

For many purposes of instruction, a simple list for core elementary transcendental
functions is given as $f \in \{\exp,\ln, \sin, \cos, \tan\}$.