General Types of Functions

Significance

There are several general classification of functions which encompass many specific types but are still associated with distinct behaviors. The ones we discuss here are

Even and Odd functions and
Algebraic and Transcendental functions

Even and Odd Functions

A function f(x) defined for all real x is called even if f(-x) = f(x) for all real x and is called odd if f(-x) = -f(x) for all real x.

Graphically, a function is even if it is symmetric about y-axis and odd if it is symmetric about the line y = -x.

Here is a list of common even and odd functions:

EVEN ODD NEITHER

xⁿ with n even xⁿ with n odd e^x

Polynomials with only even degree terms Polynomials with only odd degree terms General Polynomials

cos(x), sec(x) sin(x), csc(x) [tan(x),cot(x) on their domains] Shifted trigonometric functions such as sin(3 + x).

Algebraic and Transcendental Functions

Given a basic set of functions we can build new functions using the algebraic operations familiar from the real numbers:

Addition, Subtraction, Multiplication, and Division.
Forming negations and reciprocals.
Taking powers and extracting roots.

(See the section on algebraic operations on functions.) For example, a polynomial function can be viewed as the result of applying the operations of addition and multiplication to the set consisting of constant functions and the identity function y = x. We will apply this point of view in Stage 4 when discussing the continuity of polynomials.

Functions which can be built up starting with the constant functions and the identity function using ALL of the operations listed above are called algebraic functions.

The following are all algebraic functions:

sqrt( 1 + x^1/3),

sqrt(x³⁰ - sqrt(1 + x))

(x + x⁹)^(1/27)

(1 - x)^-4/5

whereas the following are not:

e^x,

ln(sqrt(x² + 8)),

sin(3 x² - 5)

Functions which are not algebraic are called transcendental functions.

Importance of Transcendental Functions

The basic transcendental functions - exponential, logarithmic, trigonometric, and inverse trigonometric functions - are there for a reason. Which reason depends on your level of mathematical sophistication and how much calculus you have studied. In addition to their geometric and arithmetic properties, these transcendental functions are all solutions to "differential equations" -- equations relating the values of a function to its rate of change, the rate of change of the rate of change, and so on. These will be mentioned in Stage 10.

Growth of Algebraic Functions

Algebraic functions are considerably more complicated than rational functions but share some general characteristics. In the discussion of asymptotes of rational functions we pointed out indirectly that every rational function is asymptotic to a polynomial - the one obtained by dividing through by the larges power of the variable in the denominator and setting all resulting negative powers to zero. This limits the growth of rational functions to "polynomial growth".

Algebraic functions are limited to growing like power functions.

Transcendental functions, on the other hand, can grow much faster (as in exponential functions) or much slower (as in logarithmic functions) than any power function, can be periodic (as in trigonometric functions), and can in general exhibit behavior beyond the algebraic realm.