Algebraic function

Definition

An algebraic function is a type of function defined as the root of a polynomial equation with coefficients in the real or complex numbers. It can be expressed using a finite number of operations involving addition, subtraction, multiplication, division, and taking roots.

5 Must Know Facts For Your Next Test

1. Algebraic functions can include polynomial functions, rational functions, and root functions.
2. The general form of an algebraic function is given by $P(y) = Q(x)$ where $P$ and $Q$ are polynomials.
3. Not all algebraic functions are polynomials; for example, $\sqrt{x}$ is an algebraic function but not a polynomial.
4. Algebraic functions are closed under addition, subtraction, multiplication, and composition but not necessarily under division unless the denominator is non-zero.
5. Finding the inverse of an algebraic function may involve solving polynomial equations.

Review Questions

• What distinguishes an algebraic function from a purely polynomial function?
• Provide an example of an algebraic function that is not a polynomial.
• Explain why $\sqrt{x}$ is considered an algebraic function.

Related terms

Polynomial Function: A special class of algebraic functions characterized by expressions involving only non-negative integer powers of x.

Rational Function: A ratio of two polynomial functions where the denominator is not zero.

Root Function: $f(x) = x^{1/n}$ where $n$ is a positive integer. These are also considered algebraic functions.