19 Nov 2009

Polynomials and Algebraic Functions Continued, in Edwards & Penney

presentation of Edwards & Penney's work, by Corry Shores
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[Your author is not a mathematician, he's just an admirer of Edwards & Penney's book. Please consult the text to be certain about anything in the summary below.]

Edwards & Penney's Calculus is an incredibly-impressive, comprehensive, and understandable book. I highly recommend it.

1.3 Polynomials and Algebraic Functions (continued)

Rational Functions

A rational number is a quotient of two integers. Rational functions are similar. Let's consider the form of a rational function. We will deal with two polynomials: p(x) and q(x).

Now let's consider the graphs for rational functions. They share some features with graphs for polynomials.

1) Both have a finite number of zeros. [Note first that the number of times that the graph of a polynomial crosses the x-axis is the number of zeros that it has.

And recall how a polynomial has no more zeros than its degree. Now, a fraction can equal zero only when the numerator is zero. In this case, the numerator is a polynomial, which always has a finite number of zeros. Hence, the rational function always has a finite number of zeros.]

2) The graphs for both rational and polynomial functions have a finite number of bends.

There is another interesting possibility. Consider the denominator in our equation,

It is possible for it to have a value that asymptotically approaches zero. In this case, the f(x) will be a very large number. This distinguishes rational functions from polynomials. For, the graph of a polynomial cannot have an asymptote.


Consider this equation:

Now, let's say we want to find the zero's in the numerator. We would obtain x = -2 and x = 1. And let's consider the x-intercepts (where the line meets the x-axis). Look now at the graph for the equation.

We see that the lines cross the x-axis also at -2 and 1. What are the zeros for the denominator? x = 0, x = -1, and x = 2. Now look at the asymptotes in the graph. The are found at the vertical lines at x = -1, x = 0, and x = 2. These correspond to the denominator's zeros.

Algebraic Functions

Let's consider if we begin with a power function. Then, we apply to it one of the following algebraic operations:

a) addition
b) subtraction
c) multiplication by a real number
d) multiplication
e) division, and/or
f) root-taking.

When we do so, we obtain an algebraic function. [Polynomials and rational functions are both obtained by performing algebraic operations. Hence, polynomials and rational functions are algebraic functions.]

[Polynomials are not asymptotic. So they extend across all the x-values. Rational functions, however, can have asymptotes. Hence,] a polynomial is defined everywhere across the real line. And a rational function is defined everywhere except its asymptotic zeros. However, algebraic functions can have very limited domains. Let's consider these equations along with their graphs.

The first equation has a bounded interval, but the second one has an unbounded interval. Recall graphs for polynomial and rational functions.

We notice that the graphs for the polynomial and rational functions are "smooth" at every part. But the algebraic functions can have "corners" or sharp "cusps" where the line does not appear smooth. Consider these other examples.

from Edwards & Penney: Calculus. New Jersey: Prentice Hall, 2002, pp.27-31.

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