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Transcendental Functions & Algebraic Functions: Simple Definition

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

  1. What is an Algebraic Function?
  2. What is a Transcendental Function?

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) = ax2 + bx + c . Every subtype of polynomial functions are also algebraic functions, including:
  • Power functions: any function of the form f(x) = axp, 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?

examples of transcendental functions

Most mathematicians would agree that the functions on this list are examples of transcendental functions, only because they aren’t algebraic.

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 transcendental 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:

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.


Dr. Math (2001). Defining Transcendental Functions. Retrieved May 22, 2019 from:
Larson, R. & Edwards, B. Calculus: Early Transcendental Functions. Cengage Learning. Retrieved May 22, 2019 from:

Stephanie Glen. "Transcendental Functions & Algebraic Functions: Simple Definition" From Calculus for the rest of us!

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