Algebraic and Non-Algebraic Functions

A function is called an algebraic function if it can be constructed using algebraic operations (such as addition, subtraction, multiplication, division, and taking roots) on polynomials.

Any rational function is automatically an algebraic function.

For example, `f(x)=sqrt(1/x)` , `f(x)=(x^2+1/x)/(root(3)(x^2+1/x^2)+x^4+1)` .

Functions that are not algebraic are called transcedental.

The set of transcendental functions includes the trigonometric, inverse trigonometric, exponential, and logarithmic functions, but it also includes a vast number of other functions that have never been named (for example, functions that are defined as sum of infinite series).