Rational

$\newenvironment {prompt}{}{} \newcommand {\accordion }[0]{} \newcommand {\ungraded }[0]{} \newcommand {\xmDefaultPreamble }[0]{xmPreamble.tex} \newcommand {\xmDefaultPrintstyle }[0]{xmPrintstyle.sty} \newcommand {\todo }[0]{} \newcommand {\oiint }[0]{{\large \bigcirc }\kern -1.56em\iint } \newcommand {\mooculus }[0]{\textsf {\textbf {MOOC}\textnormal {\textsf {ULUS}}}} \newcommand {\sagemath }[0]{\textsf {SageMath}} \newcommand {\surfaceColor }[0]{violet} \newcommand {\surfaceColorTwo }[0]{redyellow} \newcommand {\sliceColor }[0]{greenyellow} \newcommand {\vector }[1]{\left \langle #1\right \rangle } \newcommand {\sectionOutcomes }[0]{}$

fractions of polynomials

Rational functions are fractions of polynomials

Rational Functions

A rational function is any function that can be represented as a quotient of two polynomials

\[ \frac { a_n x^n + a_{n-1} x^{n-1} + \cdots + a_3 x^3 + a_2 x^2 + a_1 x + a_0 } { b_m x^m + b_{m-1} x^{m-1} + \cdots + b_3 x^3 + b_2 x^2 + b_1 x + b_0 } \]

where the $a_k$ and $b_k$ are real numbers and $a_n \ne 0$ and $b_m \ne 0$.

Again, we prefer polynomials in factored form. Therefore, usually, our first step is to transform the formula for the rational function to look like

\[ \frac { (x-r_n)(x-r_{n-1}) \cdots (x-r_2)(x-r_1) } { (x-s_m)(x-s_{m-1}) \cdots (x-s_2)(x-s_1) } \]

And, again, we will be able to obtain a product of exclusively linear factors with the addition of complex numbers. With real numbers we can get these products to consist of a mixture of linear and irreducible quadratics.

The graph of $y = H(w) = \frac {(w-1)}{(w+3) (w-4)} $

The graph of $y = v(r) = \frac {(r-1)}{(r+3) (r-4)^2} $

While polynomials only combined factors with positive integer powers, rational functions include factors with negative integer powers.

We can pull $H(w)$ apart to see these power-like functions.

\[ H(w) = \frac {(w-1)}{(w+3) (w-4)} = \frac {4}{7} \cdot \frac {1}{w+3} + \frac {3}{7} \cdot \frac {1}{w-4} = \frac {4}{7} (w+3)^{-1} + \frac {3}{7} (w-4)^{-1} \]

Strictly speaking, these are not power functions, but more like shifted power functions - in the denominator.

Continuity

Rational functions are continuous everywhere on their domain. They have singularities at zeros of their denominators.

Graphs of rational functions might have vertical asymptotes at these singularities, where the function grows (positively or negatively) without bound. As we can see in the examples above, the sign of the function can switch across a singularity, or not. This has to do with the power of the corresponding factor, which we will investigate later.

The graph of $y = B(c) = \frac {(c-1)(c-5)}{5(c+3)} $

In addition to horizontal and vertical asymptotes, graphs of rational functions can have oblique asymptotes, which describe the end-behavior.

The graph of a rational function need not have an asymptote at a zero of the denominator.

The graph of $y = h(t) = \frac {(t-1)(t-5)}{(t-5)(t+3)} $

$5$ is a zero of the polynomial in the denominator. However, $t=5$ is not an asymptote in the graph.

This is because $5$ is also a zero of the polynomial in the numerator (with the same exponent).

\[ y = h(t) = \frac {(t-1)(t-5)}{(t-5)(t+3)} = \frac {(t-1)}{(t+3)} \, \text { for } \, t \ne 5 \]

End-Behavior

The end-behavior of a rational function depends on the leading terms of the numerator and denominator.

$\blacktriangleright $ If $n > m$

\[ \lim \limits _{x \to \pm \infty } \frac { a_n x^n + a_{n-1} x^{n-1} + \cdots + a_3 x^3 + a_2 x^2 + a_1 x + a_0 } { b_m x^m + b_{m-1} x^{m-1} + \cdots + b_3 x^3 + b_2 x^2 + b_1 x + b_0 } = \lim \limits _{x \to \pm \infty } \frac { a_n x^n } { b_m x^m } = \pm \infty \]

$\blacktriangleright $ If $n < m$

$\blacktriangleright $ If $n = m$

ooooo-=-=-=-ooOoo-=-=-=-ooooo
more examples can be found by following this link
More Examples of Elementary Functions

2025-01-07 00:41:11