What is Rational functions: Definition and 70 Discussions

In mathematics, a rational function is any function that can be defined by a rational fraction, which is an algebraic fraction such that both the numerator and the denominator are polynomials. The coefficients of the polynomials need not be rational numbers; they may be taken in any field K. In this case, one speaks of a rational function and a rational fraction over K. The values of the variables may be taken in any field L containing K. Then the domain of the function is the set of the values of the variables for which the denominator is not zero, and the codomain is L.
The set of rational functions over a field K is a field, the field of fractions of the ring of the polynomial functions over K.

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  1. brotherbobby

    Possible values an expression can take : ##\dfrac{x^2-x-6}{x-3}##

    Problem statement : Let me copy and paste the problem as it appears in the text : Attempt 1 (from text) : The book and me independently could solve this problem. I copy and paste the solution from the book below. Attempt 2 (my own) : The problem should afford a solution using the second idea...
  2. F

    I Asymptotes of Rational Functions....

    Hello, I know that functions can have or not asymptotes. Polynomials have none. In the case of a rational functions, if the numerator degree > denominator degree by one unit, the rational function will have a) one slant asymptote and b) NO horizontal asymptotes, c) possibly several vertical...
  3. S

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    I tried graphing the function in the calculator, and the graph seems to have a horizontal asymptote at y=0, not at y=1. Why is this so? Thanks for helping out.
  4. S

    I Rational functions in one indeterminate - useful concept?

    The examples of "formal" power series and polynomials in one indeterminate are familiar and useful in algebra. However, I don't recall the example of rational functions (ratios of polynomials) in one indeterminate being used for anything. Is that concept useful? - or trivial? -or equivalent...
  5. M

    MHB What are the steps for finding asymptotes of rational functions?

    Hello everyone. Time to get back to math. I have forgotten how to find asymptotes of rational functions. I think there are three types of asymptotes. Can someone show me how to find asymptotes of rational functions? What exactly is an asymptote?
  6. Schaus

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  7. Vinay080

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  8. Math Amateur

    MHB Rational Functions - Polynomials Over a Field - Rotman Proposition 3.70

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  9. C

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  10. C

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  11. Mr Davis 97

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  12. hgducharme

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  13. B

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  14. Y

    MHB Limits of Rational Functions: Dividing by Highest Power?

    Hello all I have a general question. When I look for a limit of a rational function, there is this rule of dividing each term by the highest power. I wanted to ask if I should divide by the highest power, or the highest power in the denominator, and why ? I have seen different answers in...
  15. H

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  16. A

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  17. E

    MHB Simplify (Adding and Subtracting Rational Functions)

    5c) Simplify. \frac{2x}{3y} - \frac{x^2}{4y^3} + \frac{3}{5y^4} This is what I did, which is wrong according to the textbook. Could someone point out what I did wrong and how to correct it? Thanks. \frac{(2x)(4y^3)-(x^2)(3y)}{(3y)(4y^3)} + \frac{3}{5y^4} \frac{8xy^3-3x^2y}{(12y^3)} +...
  18. A

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  19. L

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  20. T

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  21. R

    MHB Integration of Rational Functions by Partial Fractions

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  22. I

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  23. H

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  24. L

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  25. A

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  26. J

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  27. vrmuth

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  28. V

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  29. M

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  30. T

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  31. B

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  32. J

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  33. Biosyn

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  34. M

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  35. F

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    Homework Statement Evaluate the integral. (Remember to use ln |u| where appropriate.) ∫ds/s^2(s − 1)^2 Homework Equations The Attempt at a Solution I attempted a solution using the method of partial fractions, but it seems my answer is wrong. Here's what I did... 1=A/s...
  36. F

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  37. D

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  38. D

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  39. A

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  40. V

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  41. S

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  42. B

    Integration of Rational Functions by Partial Fractions?

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  43. T

    Show that the field of rational functions is not a complete ordered field

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  44. J

    How Can I Graph Rational Functions More Systematically?

    Ok so, This summer I will be taking a Pre-calc/trig course intensive, to get ready to take calculus in the fall, to start up my track for physics. I got a Pre Calculus Workbook For Dummies and I have to say so far I'm not too pleased. I have already found a bunch of typos, and when there...
  45. Z

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  46. N

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  47. B

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  48. A

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  49. 3

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  50. N

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