General Types of Functions
SignificanceThere 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 FunctionsA 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
Here is a list of common even and odd functions:
Algebraic and Transcendental FunctionsGiven a basic set of functions we can build new functions using the algebraic operations familiar from the real numbers:
The following are all algebraic functions:
Importance of Transcendental FunctionsThe 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 FunctionsAlgebraic 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.