1. What an Algebraic Function Means
An algebraic function is any function that is formed using algebraic operations: addition, subtraction, multiplication, division, and taking roots (square roots, cube roots, etc.).
In short, if the function can be written using a finite number of algebraic steps, it is an algebraic function.
2. Formal Description
A function f is algebraic if it satisfies a polynomial equation in x and f(x). That means f(x) appears in an equation involving powers and coefficients, without trigonometric or exponential terms.
2.1. General Form
P(x, f(x)) = 0
\text{where } P \text{ is a polynomial expression in } x \text{ and } f(x)
3. Examples of Algebraic Functions
Here are common algebraic functions:
3.1. Example 1 — Polynomial Functions
All polynomial functions are algebraic.
f(x) = x^3 - 4x + 7
3.2. Example 2 — Rational Functions
Ratios of polynomials are algebraic:
f(x) = \frac{x+1}{x^2 - 5}
3.3. Example 3 — Functions with Roots
Any function involving square roots, cube roots, etc., is algebraic:
f(x) = \sqrt{x + 2}
g(x) = \sqrt[3]{x^2 - 7}
3.4. Example 4 — Combinations of Algebraic Forms
Mixing algebraic expressions still gives an algebraic function:
f(x) = x^2 + \sqrt{3x - 1}
4. What Is Not an Algebraic Function
A function is not algebraic if it includes operations beyond algebraic ones. These functions are called transcendental.
4.1. Examples of Non-Algebraic Functions
- Exponential functions: \( e^x \)
- Logarithmic functions: \( \ln x \)
- Trigonometric functions: \( \sin x, \cos x \)
- Inverse trigonometric functions: \( \arctan x \)
These cannot be written using only algebraic operations.
5. Basic Properties of Algebraic Functions
Algebraic functions behave predictably because they are built from polynomials and roots.
5.1. 1. Defined by Equations
Each algebraic function satisfies a polynomial equation in x and y.
5.2. 2. Often Have Restricted Domains
Roots and denominators can restrict which inputs are allowed.
5.3. 3. Continuous Except at Problem Points
They are usually smooth except where denominators or root conditions cause breaks.
6. Why Algebraic Functions Matter
Algebraic functions include many of the basic functions used in algebra, geometry, and calculus. They are useful because their behaviour can be understood through algebraic manipulation, making them easier to analyze compared to more complex transcendental functions.