Section OAF Other Algebraic Functions

Other Algebraic Functions

Introduction: Of all the functions that are studied, after linear and quadratic functions, there are two common class of functions that involve only the simple operations of addition, subtraction, multiplication and division. These are the classes of polynomial functions and rational functions.

The polynomial functions are characterized by the operations: finding whole number powers of a number. multiplication by a constant and addition of a constant or adding numbers combined by the previous list of operations. The rational functions are characterized by allowing further the operation of division of  numbers that result from two polynomial functions as long as that is proper. [No division is allowed by $0$.]

For  example, the function defined by assigning the value $f(x) =x^3 - 4 x $ to the number $x$, also represented symbolically as $f: x \rightarrow x^3 - 4 x$  ,  or verbally  by saying "to find $f(x)$, cube $x$, then subtract $4x$ from the result". $f$ is described as a cubic polynomial function. This  function is visualized  by the  mapping diagram, OA1.

Mapping Diagram for f(x) = x^3
                - x
Mapping Diagram OA1

For  a simple example of a rational function, we can consider the function defined by assigning the value $f(x) = \frac 2 x$ to the number $x \ne 0$, also represented symbolically as $f: x \rightarrow \frac 2 x $  ,  or verbally  by saying "to find $f(x)$, find the quotient of $2$ divided by $x$". $f$ is described as a rational function. This  function is visualized  by the  mapping diagram, OA2. Notice the arrow from $x=0$ does not arrive at the target line  to indicate the function is not defined at $x=0$.

Mapping Diagram for f(x) =
Mapping Diagram OA2

This section will provide examples, explanations, exercises and problems that will help students use the power of the mapping diagram  along with the three other tools (equations, tables , and graphs) to understand these other algebraic functions and their inverses.

Polynomial Function Definition

Rational Function Definition

One very important result about polynomial is most useful throughout the analysis of algebraic functions. Though usually stated sometime during a second course in algebra, it is not proven until possibly in some upper division college course. We state it here in a form connected to polynomial functions:

OAF.FTAThe Fundamental Theorem of Algebra

OAF.OAFI Polynomial and Rational Functions are Important. (Not yet ready.)

Example OAF.RF.0  The First Rational Function Example.

This example presents the rational (linear fractional) function $f(x) = \frac {x+1} {x-2} = 1 + \frac 3 {x-2}$ with a table of data,  a graph and a mapping diagram.

 [We have already seen examples of 2nd degree polynomial (quadratic) functions.]

Treatment of polynomial and rational functions and their graphical interpretation along with cubic and simple rational equations are often treated in algebra courses in preparation for calculus. [See  Cubic function. Rational_function]
What is missing is a balanced treatment using mapping diagrams to reinforce the function aspect of visualization.  That will be emphasis of this section. 

Comparisons will be made when appropriate to graphs- but we will develop the basic concepts for polynomial and rational functions with mapping diagrams. The end of this section includes some powerful and different ways to think about some of the most common of these functions and the ways they are represented algebraically.

Subsection OAF.PFF Polynomial Functions Forms: Roots and Factors
Subsection OAF.RFF Rational Functions Forms: Roots, Poles and Factors

Subsection OAF.CPPF Core Positive Power Functions- $x^n$ where $n>0$.

Subsection OAF.CNPF Core Negative Power Functions- $x^n$ where $n<0$.

Subsection OAF.COMP Composition and Algebraic Functions

Subsection OAF.BRF  (Asymptotic and Other) Behavior for Rational Functions

Subsection OW.ICPPF Inverse for Core Positive Power Functions- $\sqrt[n] x$ where $n>0$.
[From Section OW: Other Ways to Define Functions]

Subsection OAF.SAE Solving Equations for Algebraic Functions

Subsection OAF.APP Algebraic Function Applications (not yet ready)