# 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. ### 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. ### 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. ### Question about asymptotes of rational function

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. ### 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. ### 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. ### Graphing Rational Functions: How to Find Asymptotes and Intercepts

Homework Statement Sketch the graphs of the following functions and show all asymptotes with a dotted line y = (2x - 6)/ (x2-5x+4) i) Equation of any vertical asymptote(s) ii) State any restrictions or non-permissible value(s) iii) Determine coordinates of any intercept(s) iv) Describe the...
7. ### Who invented differential calculus for rational functions?

Euler mentions in his preface of the book "Foundations of Differential Calculus" (Translated version of Blanton): I don't understand here, who/who all had invented/discovered the study-of-ultimate ratio (differential calculus) for rational functions long before (Newton and Leibniz), without...
8. ### MHB Rational Functions - Polynomials Over a Field - Rotman Proposition 3.70

I am reading Joseph J. Rotman's book: A First Course in Abstract Algebra with Applications (Third Edition) ... I am currently focused on Section 3.5 From Numbers to Polynomials ... I need help with an aspect of the proof of Lemma 3.70 ... The relevant text from Rotman's book is as follows:In...
9. ### How to Solve a Rational Equation with Unknown Vertical Stretch/Compression?

Homework Statement Homework Equations y = f(x) y=k(x+4)(x)(x-6) y=1/f(x) y= 1/ (k(x+4)(x)(x-6)) The Attempt at a Solution I'm more looking for clarification on how people would approach this. There is no explicit point given to deduce the value of k to determine the vertical stretch or...
10. ### Rational functions: combine and simplify terms

Homework Statement (4a/a+4)+(a+2/2a) Homework Equations Just combine and then factor out The Attempt at a Solution It's actually fairly simple, but I'm having difficulty at the end. /multiply each term by opposite denominator 4a(2a)/a+4(2a) + a+2(a+4)/2a(a+4) /combine 4a(2a)+(a+2)(a+4) /...
11. ### Justification for cancellation in rational functions

For example, say we have ##\frac{x^4(x - 1)}{x^2}##. The function is undefined at 0, but if we cancel the x's, we get a new function that is defined at 0. So, in this case, we have ##x^2(x - 1)##, then ##x^2(x - 1)(1)##, and since ##\frac{x^2}{x^2} = 1##, we then have ##\frac{x^4(x - 1)}{x^2}##...
12. ### How to determine a hole in a graph?

I'm aware that in order to find the hole in a graph, you need to factor both the numerator and denominator, and look for terms that cancel out. However, is it merely just looking for a term that cancels out, or is it more specifically a term that cancels out and makes the numerator equal to...
13. ### Pre-Calc. Question: Graphing Rational Functions

When you have a rational function, such as: 3x-5/x-1 After attaining things like the x and y intercepts and asymptotes, how do you know how many "pieces" of the graph there are? With linear functions/equations, you know it's a single line. Even quadratic graphs are a single piece - albeit...
14. ### 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. ### Adv. Functions: Rate of Change in Rational Functions

Homework Statement mtan for f(x) = 5/√ 3x ... at x=1 Homework Equations msec = y2-y1 / x2-x1 The Attempt at a Solution The two points I got from the equation: (1, 5/√ 3) and (1+h, 5/√ 3+h) msec = f(1+h) - f(1) / h = (5/√ 3+h - 5/√ 3) / h ... multiply top and bottom by denominators (√ 3+h)...
16. ### Understanding limits of rational functions at infinity

Is there a way to distinguish between rational functions that have the same limit at both ends and those that don't? I think I might have answered my own question, but let's say I evaluate a rational function, and it turns out to be a coefficient ratio with no variables (3/2). Does that mean...
17. ### 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. ### Rational functions and link with direct substitution property

Homework Statement Hello, I know the direct substitution property in calculus. But, the definition of a rational function still confuses me. For example, are these rational functions: y=(x^2+2x+1)/(x+1) y=((x^2+2)^(1/2))/(x+1) The denominator of the first one could cancel. So...
19. ### Exploring Indeterminate Limits of Rational Functions at Infinity

Hi, I am in a first semester Calculus I course in college with an intermediate skill level with precalc and a basic understanding of limits and infinity. I do not understand how to solve this problem I attempted to do so only to find out after completion that ∞/∞ is indeterminate rendering my...
20. ### Laurent Series of Rational Functions

In general I am rather confused by this type of problem. The textbook has a single example and does not show (m)any of its steps so I'm lost. I have a test this coming Thursday and the following is the only question of this type that the prof. has recommended: "23. Use equations (12) and (13)...
21. ### MHB Integration of Rational Functions by Partial Fractions

1) integral (upper bound:1, lower bound:0) (x^2+1)/(x^3+x^2+4x) dx 2) integral (upper bound:1, lower bound:0) (x^4+x^2+1)/(x^3+x^2+x-3) dx Now I know how to use Partial Fractions,My question is: 1) For the first part ln(x) is not defined at 0 ¼ʃ1/x dx + ¼ʃ(3x-1)/(x²+x+4) dx = ¼ ln|x| +...
22. ### Complex Analysis - Rational Functions

Homework Statement I'm studying for my final exam and came across this problem: Let f and g be entire analytic functions and |f(z)|<|g(z)| when |z|>1. Show that f/g is a rational function. The Attempt at a Solution I really have no clue where to go :(
23. ### Singularities of two variables rational functions

Homework Statement Let p(x,y) be a positive polynomial of degree n ,p(x,y)=0 only at the origin.Is it possible that the quotient p(x,y)/[absolute value(x)+absval(y)]^n will have a positive lower bound in the punctured rectangle [-1,1]x[-1,1]-{(0,0)}? Homework Equations The Attempt at a...
24. ### Understanding horizontal asymptotes of non-even rational functions

I think the technique that is to be used for these types of problems, but I just am having trouble grasping why it is permitted. I have no problem with any homework, but it just doesn't seem right. Maybe my text is just not being clear (Larson, 9th, btw) Given a function...
25. ### MHB Polynomial and Rational Functions

For the years 1998-2009, the number of applicants to US medical schools can be closely approximated by: A(t)= -6.7615t4+114.87t3-240.1t3-2129t2+40,966 where t is the number of years since 1998. a) graph the number of applicants on 0<= t <= 11 b) based on the graph in part a, during what...
26. ### Cubic asymptotes in rational functions?

I just got this assignment for math and the question was is it possible to have a cubic asymptote in a rational function. If so explain how and where.
27. ### Finding the range of rational functions algebraically

how to find the range of rational functions like f(x) = \frac{1}{{x}^{2}-4} algebraically , i graphed it and seen that (-1/4,0] can not be in range . generally i am interested in how to find the range of functions and rational functions in particular
28. ### Limit Proof with Rational Functions

Homework Statement If r is a rational function, use Exercise 57 to show that ##\mathop {\lim }\limits_{x \to a} \space r(x) = r(a)## for every number a in the domain of r. Exercise 57 in this book is: if p is a polynomial, show that ##\mathop {\lim }\limits_{x \to a} \space p(x) = p(a)##...
29. ### Integrating Rational Functions

1. ∫\frac{dx}{x3 + 2x} We're suppose to evaluate the integral. Use Partial Fraction Decomposition: \frac{1}{x3 + 2x} = \frac{A}{x} + \frac{Bx + C}{x2 + 2} 1 = A(x2 + 2) + (Bx + C)(x) 1 = Ax2 + 2A + Bx2 + Cx 1 = x2( A + B) + Cx + 2A Solving for A gives \frac{1}{2} Solving for B...
30. ### Rational functions w/ common factors don't simplify?

Say we have a rational function P(x)=(x^2-3x-4)/(x-4)=[(x+1)(x-4)]/(x-4) I'm a little confused as to why the (x-4) doesn't cancel out. It graphs the same as y=x+1 for x≠4. I feel like I'm missing something from the order of operations.
31. ### F-automorphism group of the field of rational functions

I've been doing some exercises in introductory Galois theory (self-study hence PF is the only avaliable validator :) ) and a side-result of some of them is surprising to me, hence I would like you to set me straight on this one if I'm wrong. Homework Statement Let K(x) be the field of rational...
32. ### Integration of Rational Functions by Partial Fractions

Homework Statement ∫(x3+4)/(x2+4)dx Homework Equations n/a The Attempt at a Solution I know I have to do long division before I can break this one up into partial fractions. So I x3+4 by x2+4 and got x with a remainder of -4x+4 to be written as x+(4-4x/x2+4). Then I rewrote...
33. ### Integration of Rational Functions

Homework Statement ∫(7x^2+22x-54)/((x-2)(x+4)(x-1)) dx Homework Equations Partial functions The Attempt at a Solution ∫(7x^2+22x-54)/((x-2)(x+4)(x-1)) dx = ∫(Ax)/(x2+2x-8) + B/(x-1) = ∫A(x-1)dx + ∫B(x^2+2x-8)dx Or is it...
34. ### Integrating rational functions

Question: When determining the coefficients of the partial fractions for say 5 or more coefficients... Do you find it easiest to set up linear equations and solving? Any advice would be appreciated... Next question.. look in paint doc... why would I3 not be equal to I21??
35. ### Integration of Rational Functions

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. ### Integration of Rational Functions by Partial Fractions

Homework Statement Evaluate the integral. (Remember to use ln |u| where appropriate.) ∫(x^3 + 36)/(x^2 + 36) Homework Equations The Attempt at a Solution A little bit confused about arriving at the solution for this problem. I get stuck a little ways in. Any help would be...
37. ### Integrating Log and Rational Functions

Homework Statement \int_0^{\infty} \ln \left( \frac{e^x+1}{e^x-1} \right) \mbox{d}x \int_0^{\infty} \frac{1}{x^n+1}\ \mbox{d}x\ \forall n >1 Homework Equations - The Attempt at a Solution I've tried IBP and separating the ln into two terms and failed. I've also tried a...
38. ### Basic limits of rational functions: behavior near vertical asymptotes

Homework Statement We are required to sketch a (reasonably accurate) picture of a rational function f(x) = P(x)/Q(x) with P, Q polynomials in x and Q nonzero. We know that the roots of Q(x) are, say, x1, x2, etc. and so f(x) is (typically) asymptotic to the vertical lines x = xk for each k...
39. ### Evaluating limits of rational functions

Homework Statement Why does the limit as x approaches 0 of x^2 + 5 / 3x go to infinity (with 0 as an essential disc.) but without the +5, the function goes to 0? Homework Equations The Attempt at a Solution I tried approaching evaluating the limit of the function by comparing the...
40. ### Integration of rational functions by partial fractions

Homework Statement write out the form of the partial fraction decomposition of the function, do not determine the numerical values of the coefficients x^2/(x^2 + x + 2) Homework Equations The Attempt at a Solution since the numerator is not less of a degree than the...
41. ### Domains of Rational Functions (standard notation)

Im preparing for a CLEP test in precalculus. As part of my prep, I need to review identifying domains of functions. I have a question about writing domains in standard notation. I was hoping someone could explain a bit the style. For an example: x-2 / x^2 -2x -35 As a rational...
42. ### Integration of Rational Functions by Partial Fractions?

Integration of Rational Functions by Partial Fractions? Ok I'm working on some homework problems and I don't even know how to do the first one, here is my problems and the steps that I did thus far ( I don't know if I did them right) 5x-13/(x-3)(x-2)= A/x-3 + B/x-2\rightarrow ...5x-13=...
43. ### Show that the field of rational functions is not a complete ordered field

Homework Statement Show that R(x) cannot be made into a complete ordered field, where R(x) is the field of rational functions. Homework Equations Definition of a complete ordered field: An ordered field O is called complete if supS exists for every non empty subset S of O that is...
44. ### 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. ### Rational Functions – Homework Help

Hi there I have a Rational function y = 1 / x^2-1 . I have a good idea what the graph looks it, it will have vertical asymptotes at -1 and 1 and I can work out the y intercept (-1 concave down). However I'm not sure about the other parts to the question. Homework Statement dy/dx...
46. ### How to state an equation of rational functions that has Asymptotes?

How to state the equations of a rational functions with the following asymptotes? (1)x=2, y=-3 (2)y=0, x=4 (3)y=0
47. ### Help in need : Rational functions problem

Homework Statement A scientist predicted that the population of fish in a lake could be modeled by the function f(t)= 40t/(t^2+1), where t is given in days. The function that actually models the fish population is g(t)=45t/(t^2+8t+7). Determine where g(t)>f(t). Homework Equations...
48. ### Integration of rational functions by partial fractions

The problem is: ((x^3)+x)/(x-1) And i need to break it into partial fractions... I tried long division and got: ((x^2) +x ) But the book gives me the answer of: (x^2)+x+2+(2/(x-1)) Any help would be very much appreciated, thanks.
49. ### Simplifying rational functions with common factors

Homework Statement Simply these rational functions: [\sqrt{(X^2)+12}-4]/(X-2) (2-\sqrt{(X^2)-5})/(X+3) (X-1)/(\sqrt{X+3}-2) Homework Equations The only example in the book used the technique of multiplying the numerator and denominator by the function p(x) if p(x) is the function...
50. ### Rational Functions: Degree of Denominator vs Nominator

Hi all. I have always wondered: If we e.g. look at functions given by f(x) = \frac{\cos x}{x^2}, \quad g(x) = \frac{\sin x}{x^2}, \quad h(x) = \frac{\exp x}{x^2}, then does the degree of the denominator exceed the degree of the nominator by 1 or by 2?