# Questions tagged [rational-functions]

Rational functions are ratios of two polynomials, for example $(x+5)/(x^2+3)$.

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### How to solve $\int\frac{4y+3}{4y^2-9}dy$? [closed]

For an assignment from my book I have to evaluate: $$\int\frac{4y+3}{4y^2-9}dy$$ (and therefore actually solve it), but I don't know how to start. Thanks in advance for your help! For your ...
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### Existence of a rational isomorphism between circles

Reference: "The Arithmetic of Elliptic Curves" (Silverman, second edition), Exercise 1.11. (b). Consider two circles $X^2 + Y^2 = p \cdot Z^2$ and $X^2 + Y^2 = q \cdot Z^2$. If $p$ and $q$ ...
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### Determine whether the given is rational functions or rational equation

Is a rational function or a rational equation or none of these? 1.) $y=5x³-2x+1$ 2.) $g(x)=7x³-4√x+1/x²+3$ Hope you'll help me thanks
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### For rational functions, is there a way to tell from a graph alone how many powers the denominator is above the numerator? [closed]

For rational functions, is there a way to tell how many powers the denominator is above the numerator only based on the graph? I know that if the denominator has a greater degree than the numerator, ...
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### Partial Fractions With Repeated Quadratics

I'm told that given a function $f(x)=\frac{P(x)}{Q(x)}$, if $\deg(P)>\deg(Q)$ then $f$ is improper, which makes sense when I think of real numbers like $5/2$. And in this case we would have to do ...
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### Simplifying a rational function on an algebraic curve

Does a rational function $\phi$ on a smooth projective algebraic curve $F$ over a algebraically closed field $K$ always have a representative $\frac{f}{g}$, where $f$ and $g$ are polynomials without ...
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### Integration - Evaluate $\int \frac{x^7+2}{(x^2+x+1)^2} \ dx$

Evaluate $$\int \frac{x^7+2}{(x^2+x+1)^2} \ dx$$ This problem is from G N Berman, no. 2056 (integrate using ostrogradsky's method). I referred to this question as well as this article but I could ...
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I have the following equation $$xy^2-\frac{x(-xy-1)^2}{(x-1)^2}+2y\left(\frac{-xy-1}{x-1}-1\right)+\left(\frac{-xy-1}{x-1}-1\right)^2=0$$ which I was trying to simplify. I know the solution is $$\frac{... 1answer 34 views ### Why is the y intercept for this equation positive? I am tasked with sketching the following rational function$$\frac{\left(x+1\right)\left(3x+2\right)}{\left(x+1\right)^{2}}$$I have found that the two asymptotes are: y=3 and x=-1 and there are ... 2answers 74 views ### Finding values of a, b, c, d such that a f(x)=\frac{ax+d}{cx+b} is self inverse I am trying to solve the following problem: (screenshot) For which numbers a, b, c, and d will the function$$f(x)=\frac{ax+b}{cx+d}$$satisfy f(f(x))=x for all x? I have a solution, but ... 0answers 31 views ### Evaluating contour integrals over the unit circle of rational functions. Let p(z):= a_0 + a_1z + \cdots + a_nz^n be a degree n polynomial, let m be a large integer (which we may assume much larger than n), and let k be some integer in the range m+1, \cdots, m+n.... 0answers 31 views ### Rational functions with rational function antiderivatives Which rational functions over \mathbb{R} have rational functions as antiderivatives? I can think of a few trivial examples. Polynomials over \mathbb{R} are rational functions and of course have ... 3answers 62 views ### A faster way to solve \frac1{x^2+6x-15}+\frac2{x^2+6x+3}=\frac3{x^2+6x+1} in a timed test? What is a faster way to do this problem? Solve the equation$$ \frac{1}{x^2+6x-15} + \frac{2}{x^2+6x+3} = \frac{3}{x^2+6x+1} $$This is from a timed test, so the fastest answers would be the best. I ... 1answer 52 views ### Why is -2 an answer for \frac{|x+3|+x}{x+2} > 1? In the inequality \frac{|x+3|+x}{x+2} > 1, by doing sign chart method the answer I am getting is x \in (−5,−2)\cup(−1,\infty) But the graph in Desmos shows that it should be x \in (−5,−2]\cup(... 1answer 71 views ### How to simplify this fraction \frac{x^2+7x+1}{x^2-4}? [closed] I'm familiar with polynomial long division but I keep getting stuck when I try to answer this question. Online math calculations also say there's no way to simplify this (but it's a homework question ... 0answers 29 views ### Taylor series of degenerate rational function I have a rational function f which is well-defined on (0, 1], degenerate at zero. I can find the limits as x \rightarrow 0 of f and its derivatives, but find that the third and higher ... 1answer 29 views ### Rational Function Approximation of the Maximum Function [closed] Is is possible to create a rational function R(x) that is an approximation of \max(x,0) (that is, as x \to -\infty, \, R(x) \to 0 and as x \to \infty, \, R(x) \to x)? 4answers 78 views ### Find minimum value of f(x)=\frac{(x^2-x+1)^3}{x^6-x^3+1} [closed] Find minimum value of f(x)  where$$f(x)=\frac{(x^2-x+1)^3}{x^6-x^3+1}$$On differentiating I got$$f'(x)=\frac{3(x^2-x+1)^2\left(x^6-2x^5-x^4+x^2+2x-1\right)}{(x^6-x^3+1)^2}$$which doesn't help ... 1answer 192 views ### What would the picture for partial fractions look like? As how integration by parts has the picture below, what would the picture for partial fractions look like? Although there's probably no way to escape the heavy algebra necessary for partial fractions,... 0answers 44 views ### How to approximate f(x)=\frac{a_1x^2+a_2x+a_3}{\sqrt{a_4x^2+a_5x+a_6}} + \frac{b_1x^2+b_2x+b_3}{\sqrt{b_4x^2+b_5x+b_6}}? How to approximate the following function?$$f(x) = \frac{a_1x^2+a_2x+a_3}{\sqrt{a_4x^2+a_5x+a_6}} + \frac{b_1x^2+b_2x+b_3}{\sqrt{b_4x^2+b_5x+b_6}}$$where a_i and b_i are constants. I thought ... 2answers 151 views ### Calculus of  \lim_{(x,y)\to (0,0)} \frac{8 x^2 y^3 }{x^9+y^3}  By Wolfram Alpha I know that the limit$$ \lim_{(x,y)\to (0,0)} \dfrac{8 x^2 y^3 }{x^9+y^3}=0. $$I have tried to prove that this limit is 0, by using polar coordinate, the AM–GM inequality and ... 2answers 57 views ### Bounding coefficients of the polynomial (1-x^2)^n (1-x)^{-m}. I have been trying to get some upper bound on the coefficient of x^k in the polynomial$$(1-x^2)^n (1-x)^{-m}, \text{ $m \le n$}.$$A straightforward calculation shows that for even k, the ... 0answers 31 views ### Rational Function Is Eventually Greater Than 0 Let m, n \in \mathbb{Z}^{\geq 0}. Let a_1, ..., a_n \in \mathbb{R}. Let b_1, ..., b_m \in \mathbb{R}. Assume \displaystyle\frac{a_n}{b_m} > 0. Let f(x) = \displaystyle\frac{a_n x^n + a_{n -... 1answer 46 views ### Is y=\frac{x^3+0.3x-7}{2x-1} Rational or Quotient Function? I understand this is an easy one question however, I just wanted to clarify which it would be considered as a rational function or a quotient function. I understand that a rational function has y=\... 1answer 108 views ### If a_i\in\mathbb{R}, \omega^2+\omega+1=0, and \sum_{i=1}^n\frac{1}{a_i+\omega^k} =2\omega^{2k} for k=1,2, find \sum_{i=1}^n\frac{1}{a_i+1}. In this question, \omega is the complex cube root of 1 and a_i \in \mathbb R. If$$\sum_{i=1}^n \frac{1}{a_i + \omega} =2\omega ^2$$and$$\sum_{i=1}^n \frac{1}{a_i + \omega ^2} =2\omega\,,$$... 1answer 29 views ### Intuition and Visualization of Limits of Complex Rational Functions I am having some difficulty evaluating and visualizing limits of complex rational functions. For example,$$ \lim_{z \to z_0 } f(z),\ where\ Im(z_0)\ne 0,\ Re(z_0)\ne 0f(z):=\frac{L(z)}{M(z)}, ...
Suppose I have a rational function $R(X) = P(X)/Q(X)$. For simplicity, let's assume that the roots of $Q$ are all real and simple, and $\deg P < \deg Q = n$. Then I can decompose $R$ using partial ...