rational function

Examples of rational function in the following topics:

• Introduction to Rational Functions

  • A rational function is one such that $f(x) = \frac{P(x)}{Q(x)}$, where $Q(x) \neq 0$; the domain of a rational function can be calculated.
  • A rational function is any function which can be written as the ratio of two polynomial functions.
  • Note that every polynomial function is a rational function with $Q(x) = 1$.
  • A constant function such as $f(x) = \pi$ is a rational function since constants are polynomials.
  • Factorizing the numerator and denominator of rational function helps to identify singularities of algebraic rational functions.

• Solving Problems with Rational Functions

  • Rational functions can be graphed on the coordinate plane.
  • Rational functions can have zero, one, or multiple $x$-intercepts.
  • In the case of rational functions, the $x$-intercepts exist when the numerator is equal to $0$.
  • Set the numerator of this rational function equal to zero and solve for $x$:
  • Use the numerator of a rational function to solve for its zeros

• Polynomial and Rational Functions as Models

  • Polynomial and rational functions are both relatively accurate and easy to use.
  • To deal with the asymptotic problems of polynomials, we also use rational functions:
  • A rational function is the ratio of two polynomial functions and has the following form:
  • For example, if $n=2$ and $m=1$, the function is described as a quadratic/linear rational function.
  • Polynomials and rational functions are used for approximation in many everyday devices.

• Partial Fractions

  • Partial fraction decomposition is a procedure used to reduce the degree of either the numerator or the denominator of a rational function.
  • To find a coefficient, multiply the denominator associated with it by the rational function $R(x)$:
  • We have rewritten the initial rational function in terms of partial fractions.
  • Apply decomposition to the rational function $g(x) = \frac{8x^2 + 3x - 21}{x^3 - 7x - 6}$
  • For a rational function $R(x) = \frac{f(x)}{g(x)}$, if the degree of $f(x)$ is greater than or equal to the degree of $g(x)$, the function cannot be decomposed in a straightforward way.

• Domains of Rational and Radical Functions

  • A rational expression is one which can be written as the ratio of two polynomial functions.
  • Despite being called a rational expression, neither the coefficients of the polynomials nor the values taken by the function are necessarily rational numbers.
  • The domain of a rational expression of is the set of all points for which the denominator is not zero.
  • To find the domain of a rational function, set the denominator equal to zero and solve.  
  • Calculate the domain of a rational or radical function by finding the values for which it is undefined

• The Method of Partial Fractions

  • Partial fraction expansions provide an approach to integrating a general rational function.
  • Partial fraction expansions provide an approach to integrating a general rational function.
  • Any rational function of a real variable can be written as the sum of a polynomial and a finite number of rational fractions whose denominator is the power of an irreducible polynomial and whose numerator has a degree lower than the degree of this irreducible polynomial.

• Rational Inequalities

  • Because the inequality is written as $\geq0$ as opposed to $>0$, we will need to evaluate the $x$ values at zeros to determine whether the function is defined.
  • In the case of $x=-2$ and $x=2$, the rational function has a denominator equal to zero and becomes undefined.
  • In the case of $x=-3$ and $x=1$, the rational function has a numerator equal to zero, which makes the function overall equal to zero, making it inclusive in the solution.
  • For $x$ values that are zeros for the numerator polynomial, the rational function overall is equal to zero.
  • For $x$ values that are zeros for the denominator polynomial, the rational function is undefined, with a vertical asymptote forming instead.

• The Intermediate Value Theorem

  • For each value between the bounds of a continuous function, there is at least one point where the function maps to that value.
  • It is false for the rational numbers Q.
  • However there is no rational number x such that f(x) = 0, because √2 is irrational.
  • A graph of a rational function, .
  • A discontinuity occurs when : the function is not defined at $x=-2$.

• Zeroes of Polynomial Functions With Rational Coefficients

  • A real number that is not rational is called irrational.
  • The term rational in reference to the set Q refers to the fact that a rational number represents a ratio of two integers.
  • In mathematics, the adjective rational often means that the underlying field considered is the field Q of rational numbers.
  • Rational polynomial usually, and most correctly, means a polynomial with rational coefficients, also called a "polynomial over the rationals".
  • However, rational function does not mean the underlying field is the rational numbers, and a rational algebraic curve is not an algebraic curve with rational coefficients.

• Integer Coefficients and the Rational Zeros Theorem

  • In algebra, the Rational Zero Theorem, or Rational Root Theorem, or Rational Root Test, states a constraint on rational solutions (also known as zeros, or roots) of the polynomial equation
  • Since any integer has only a finite number of divisors, the rational root theorem provides us with a finite number of candidates for rational roots.
  • The cubic function $3x^3-5x^2+5x-2$ has one real root between $0$ and $1$.
  • We can use the Rational Root Test to see whether this root is rational.
  • Use the Rational Zeros Theorem to find all possible rational roots of a polynomial