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 $f \in \{\exp,\ln, \sin, \cos, \tan\}$.