rational function

(noun)

Any function whose value can be expressed as the quotient of two polynomials (where the polynomial in the denominator is not zero).

Related Terms

  • Example
  • domain
  • singularities
  • vertical asymptote
  • asymptote
  • oblique
  • numerator
  • denominator

(noun)

Any function whose value can be expressed as the quotient of two polynomials (except division by zero).

Related Terms

  • Example
  • domain
  • singularities
  • vertical asymptote
  • asymptote
  • oblique
  • numerator
  • denominator

Examples of rational function in the following topics:

  • Introduction to Rational Functions

    • A rational function is one such that f(x)=P(x)Q(x)f(x) = \frac{P(x)}{Q(x)}f(x)=​Q(x)​​P(x)​​, where Q(x)≠0Q(x) \neq 0Q(x)≠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)=1Q(x) = 1Q(x)=1.
    • A constant function such as f(x)=πf(x) = \pif(x)=π 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 xxx-intercepts.
    • In the case of rational functions, the xxx-intercepts exist when the numerator is equal to 000.
    • Set the numerator of this rational function equal to zero and solve for xxx:
    • 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=2n=2n=2 and m=1m=1m=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)R(x)R(x):
    • We have rewritten the initial rational function in terms of partial fractions.
    • Apply decomposition to the rational function g(x)=8x2+3x−21x3−7x−6g(x) = \frac{8x^2 + 3x - 21}{x^3 - 7x - 6}g(x)=​x​3​​−7x−6​​8x​2​​+3x−21​​
    • For a rational function R(x)=f(x)g(x)R(x) = \frac{f(x)}{g(x)}R(x)=​g(x)​​f(x)​​, if the degree of f(x)f(x)f(x) is greater than or equal to the degree of g(x)g(x)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 ≥0\geq0≥0 as opposed to >0>0>0, we will need to evaluate the xxx values at zeros to determine whether the function is defined.
    • In the case of x=−2x=-2x=−2 and x=2x=2x=2, the rational function has a denominator equal to zero and becomes undefined.
    • In the case of x=−3x=-3x=−3 and x=1x=1x=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 xxx values that are zeros for the numerator polynomial, the rational function overall is equal to zero.
    • For xxx 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=−2x=-2x=−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 3x3−5x2+5x−23x^3-5x^2+5x-23x​3​​−5x​2​​+5x−2 has one real root between 000 and 111.
    • 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
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