🧮Symbolic Computation Unit 4 – Rational Function Manipulation

Key Concepts and Definitions

• Rational functions defined as the ratio of two polynomial functions $\frac{P(x)}{Q(x)}$, where $Q(x) \neq 0$
• Domain of a rational function consists of all real numbers except those that make the denominator equal to zero
• Vertical asymptotes occur when the denominator of a rational function equals zero and the numerator does not
• Horizontal asymptotes determined by comparing the degrees of the numerator and denominator polynomials
  • If degree of numerator < degree of denominator, horizontal asymptote is y = 0
  • If degree of numerator = degree of denominator, horizontal asymptote is y = leading coefficient of numerator divided by leading coefficient of denominator
  • If degree of numerator > degree of denominator, no horizontal asymptote exists
• Oblique (slant) asymptotes occur when the degree of the numerator is exactly one more than the degree of the denominator
• Zeros of a rational function are the x-intercepts where the numerator equals zero and the denominator does not
• Holes (removable discontinuities) occur when a factor cancels out between the numerator and denominator, resulting in an undefined point that can be filled in by simplifying the function

Properties of Rational Functions

• Rational functions are continuous on their domain, which excludes points where the denominator equals zero
• The graph of a rational function will have at most one horizontal or oblique asymptote
• The behavior of a rational function near a vertical asymptote depends on the sign of the leading coefficients of the numerator and denominator
  • If leading coefficients have the same sign, the function approaches positive or negative infinity on both sides of the asymptote
  • If leading coefficients have opposite signs, the function approaches positive infinity on one side and negative infinity on the other side of the asymptote
• Rational functions with a higher degree in the numerator than the denominator will have a graph that resembles a polynomial function of the same degree as the numerator
• The end behavior of a rational function is determined by the horizontal or oblique asymptote, if one exists
• Rational functions can have local maxima and minima, determined by the first derivative test or sign analysis of the function
• The multiplicity of a zero or a pole (vertical asymptote) affects the shape of the graph near that point (e.g., double zero results in a "bounce" off the x-axis)

Simplification Techniques

• Factor the numerator and denominator to identify and cancel common factors, simplifying the rational function
• Multiply the numerator and denominator by the least common denominator (LCD) to eliminate fractions within the function
• Use long division or synthetic division to divide the numerator by the denominator, rewriting the rational function as a polynomial plus a simpler rational function
• Apply the difference of squares formula $a^2 - b^2 = (a+b)(a-b)$ to factor and simplify rational functions containing squared terms
• Combine rational functions with the same denominator by adding or subtracting their numerators
• Simplify complex fractions by multiplying the numerator and denominator by the LCD of all fractions involved
• Recognize and simplify rational functions that can be reduced to a single polynomial term (e.g., $\frac{x^2 + 2x + 1}{x + 1} = x + 1$)

Operations on Rational Functions

• Addition and subtraction of rational functions involves finding the least common denominator (LCD) and adding or subtracting the numerators
  • Find the LCD by factoring each denominator and taking the product of the highest power of each factor
  • Multiply each rational function by the LCD divided by its denominator to obtain a common denominator
  • Add or subtract the resulting numerators and simplify the result
• Multiplication of rational functions involves multiplying the numerators and denominators separately, then simplifying the result
• Division of rational functions is performed by multiplying the first function by the reciprocal of the second function, then simplifying
• Composition of rational functions $(f \circ g)(x)$ is achieved by substituting the inner function $g(x)$ into the outer function $f(x)$ and simplifying
• To find the inverse of a rational function $f(x) = \frac{P(x)}{Q(x)}$, swap $x$ and $y$, then solve for $y$ in terms of $x$
  • The domain of the inverse function will be the range of the original function, and vice versa
  • The inverse of a rational function may not always be a rational function itself

Polynomial Division and Partial Fractions

• Polynomial long division used to divide a higher-degree polynomial by a lower-degree polynomial, resulting in a quotient and a remainder
  • The quotient and remainder can be used to rewrite a rational function as a polynomial plus a simpler rational function
• Synthetic division is a shortcut method for dividing a polynomial by a linear factor of the form $(x - a)$
  • The result of synthetic division is the quotient polynomial, with the remainder being the last number in the bottom row
• Partial fraction decomposition is the process of breaking down a rational function into a sum of simpler rational functions
  • Partial fractions are useful for integration and simplifying complex rational expressions
• Steps for partial fraction decomposition:
  1. Ensure the degree of the numerator is less than the degree of the denominator (perform polynomial long division if necessary)
  2. Factor the denominator completely
  3. Determine the form of the partial fractions based on the types of factors in the denominator (linear, repeated linear, irreducible quadratic)
  4. Set up a system of equations by equating the original rational function to the sum of the partial fractions with unknown coefficients
  5. Solve the system of equations to find the values of the unknown coefficients
  6. Write the final answer as a sum of the partial fractions with their respective coefficients

Graphing Rational Functions

• To graph a rational function, follow these steps:
  1. Find the domain of the function by setting the denominator equal to zero and solving for x
  2. Identify any vertical asymptotes (x-values not in the domain) and plot them as dashed vertical lines
  3. Determine the presence and equations of any horizontal or oblique asymptotes
  4. Find the x- and y-intercepts by setting the numerator equal to zero (for x-intercepts) and setting x equal to zero (for the y-intercept)
  5. Identify any holes in the graph by factoring the numerator and denominator and canceling common factors
  6. Plot additional points as needed to sketch the graph, paying attention to the behavior near asymptotes and intercepts
• Use the sign analysis method to determine the behavior of the function between vertical asymptotes and intercepts
  • Create a sign chart by placing the factors of the numerator and denominator on separate rows, identifying the signs of each factor in the intervals between zeros and asymptotes
  • Multiply the signs in each column to determine the sign of the function in each interval
• Utilize transformations to graph rational functions more efficiently
  • Horizontal shifts: Replace $x$ with $(x - h)$ to shift the graph $h$ units to the right
  • Vertical shifts: Add $k$ to the function to shift the graph $k$ units up
  • Horizontal stretches/compressions: Replace $x$ with $\frac{x}{a}$ to stretch the graph horizontally by a factor of $|a|$ (if $|a| > 1$) or compress it (if $0 < |a| < 1$)
  • Vertical stretches/compressions: Multiply the function by $a$ to stretch the graph vertically by a factor of $|a|$ (if $|a| > 1$) or compress it (if $0 < |a| < 1$)
  • Reflections: Replace $x$ with $-x$ for a reflection across the y-axis, or multiply the function by $-1$ for a reflection across the x-axis

Applications in Symbolic Computation

• Symbolic computation software (e.g., Mathematica, Maple, SymPy) can be used to manipulate and simplify rational functions
• These tools can perform operations such as factoring, expanding, simplifying, and finding asymptotes and intercepts
• Example of simplifying a rational function using SymPy:

from sympy import *

x = Symbol('x')
f = (x**2 + 3*x + 2) / (x**2 - 4)

simplify(f)

Output:

(x + 2)/(x - 2)

• Symbolic computation can also be used to find the partial fraction decomposition of a rational function:

from sympy import *

x = Symbol('x')
f = (2*x + 1) / ((x - 1)*(x + 2))

apart(f)

Output:

1/(x - 1) + 1/(x + 2)

• These tools can also compute limits, derivatives, and integrals of rational functions, which are essential for more advanced applications in calculus and engineering
• Symbolic computation enables the automation of complex algebraic manipulations, saving time and reducing the risk of errors in manual calculations

Common Pitfalls and Tips

• Remember to factor the numerator and denominator completely before attempting to simplify or perform operations on rational functions
• Be cautious when canceling factors in rational functions, ensuring that the factors are not equal to zero for the given domain
• When finding the LCD for addition or subtraction of rational functions, make sure to include the highest power of each factor from all denominators
• Avoid dividing by zero when evaluating rational functions or simplifying expressions
• When graphing rational functions, pay close attention to the behavior near vertical asymptotes and holes, as these can be easily overlooked
• Double-check the signs of the leading coefficients when determining the behavior of a rational function near a vertical asymptote
• Remember that the degree of the numerator and denominator determines the presence and type of horizontal or oblique asymptotes
• When performing partial fraction decomposition, ensure that the degree of the numerator is less than the degree of the denominator, performing polynomial long division if necessary
• Regularly practice simplifying and manipulating rational functions to develop fluency and avoid common mistakes
• Utilize symbolic computation tools to verify your results and explore more complex problems, but strive to understand the underlying concepts and techniques