Functions can be separated into two types: **algebraic functions** and **transcendental functions. **

## What is an Algebraic Function?

**Algebraic functions** are built from finite combinations of the basic algebraic operations: addition, subtraction, multiplication, division, and raising to constant powers.

Three important types of algebraic functions:

**Polynomial functions**, which are made up of monomials. The terms can be made up from constants or variables. For example, 2x + 1, xyz + 50, f(x) = ax^{2}+ bx + c . Every subtype of polynomial functions are also algebraic functions, including:*Linear functions*, which create lines and have the form y = mx + b,*Cubic functions*, third degree polynomials which have the form

*Quadratic functions*: fourth degree polynomials with the form

.

**Power functions**: any function of the form f(x) = ax^{p}, where: a and p are constants, p is a real number,

and a is nonzero.**Rational functions**: any function where one polynomial function is divided by another.

## What is a Transcendental Function?

The exact definition of what a transcendental function is is hard to pin down. That’s because it’s generally defined by exclusion:**If a function isn’t algebraic, then it’s transcendental.**

I say “generally”, because there isn’t an exact consensus, and individual authors may state a certain function is transcendental, while another may disagree. Some authors might state that these functions are generated by infinite series; While that is true, it’s also true that some algebraic functions can be generated by infinite series as well. While algebraic functions are a set of small, precisely defined functions (e.g. constant functions, exponential functions), transcendentals are simply “everything else.”

According to a 2001 post by Dr. Math, there are also several “named” transcendental functions. For example: Bessel functions and Hankel functions. These are usually only found in very specialized areas, so if you’re in an elementary calculus class, you’re unlikely to come across them.

A few examples of functions that most people agree are transcendental in nature include:

- Logarithmic Functions: Log(x).
- Exponential functions: For example, x
^{2; x21x2 + x. } - Trigonometric Functions: For example, sin(x); cos(x).
- Any function containing e
^{x}.

## Why are they called Transcendental Functions?

Transcendental, in math, means “non-algebraic”. These functions “transcend” the usual rules of algebra (*transcend *means to “go beyond the range or limits of…”). Transcendentals were first defined by Euler in his Introductio (1748) as functions not definable by the “ordinary operations of algebra”.

## Limits of Transcendental Functions

Limits of Transcendental Functions can be found with direct substitution.

## References

Dr. Math (2001). Defining Transcendental Functions. Retrieved May 22, 2019 from: http://mathforum.org/library/drmath/view/54593.html

Larson, R. & Edwards, B. Calculus: Early Transcendental Functions. Cengage Learning. Retrieved May 22, 2019 from: https://books.google.com/books?id=zUfAAgAAQBAJ

**CITE THIS AS:**

**Stephanie Glen**. "Transcendental Functions & Algebraic Functions: Simple Definition" From

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