Understand Rational Functions with Examples and Solutions

High School
Algebra
Algebra II
Rational functions

"I have encountered a question like this:
Suppose a complex function f is analytic in |z|<1, continuous in $| z | \leq 1$ , and |f|=1 on |z|=1. Show that f can be extended as a rational function.
According to the isolation property of zero points, f has only finite zero points in $| z | \leq 1$ . But I don’t know how to show that it’s a rational function. Hope someone could help.

Jupellodseple804 2022-02-15

Let X be an integral and Noetherian scheme . We consider an open subset U of X and $f ϵ Γ (U, O_{X})$ . Hence we may regard (U,f) as a rational function in X. I would like to prove that if $p ϵ X - U$ (and there is no rational function defined at p that is equivalent to (U,f)), there exists an open subset V of X containing p such that we may write the inverse rational function of (U,f) as (V,g) for certain $g ϵ Γ (V, O_{X})$ .
The idea behind this is that if a rational function has a pole at a point, then its inverse rational function has a zero, and in particular it is defined at that point.

sacateundisco8i3 2022-02-15

Let k be a field, $A = \frac{k [x, y]}{(y^{2}, x y)}$ . I know that the only associated points of Spec A are [(x,y)] and [(y)]. I want to show that
$\frac{x - 2}{(x - 1) x}$
is not a rational function.
Intuitively it makes sense to me that this function is not defined at the origin [(x,y)], hence not a rational function. But I am having trouble arguing it precisely. I would appreciate any comments. Thanks)

Olivia Vasquez 2022-02-15

I wanted to find a function f(x) that has the following properties:
1.extreme values at x=-1 and x=0 and x=1
2.x -axis is asymptote
No other asymptotes or extrema.
I found the following two by integrating and trial and error:
$f_{1} (x) = x^{2} e^{- x^{2}}$
$f_{2} (x) = 2 \ln (x^{2} + 1) + \ln (x^{4} - x^{2} + 1)$
How can I find one that is actually a rational function?

Corbin Walton 2022-02-15

Hekp me with regarding this rational function:
$f (x) = \frac{x - 3}{- x + 1}$

Adrienne Rowe 2022-02-15

Suppose that X is a varity over an algebraically close field K. If I am not mistaken, a rational function over X is a function in K(X). And a function on X is regular at a point $x \in X$ if there exists $g, h \in K [X]$ , and an open subset $V \subseteq X$ containing x, such that for all $y \in V, h (y) \neq 0$ and $f (y) = \frac{g (y)}{h (y)}$ . And f(x) is a regular function if it is regular at every point in X.
I not understand why rational functions need not be regular, and I am wondering if there is any example of rational function being not regular.

Ausanioyck 2022-02-15

Let's consider the rational functions whose numerator and denominator of the function term are coprime.
Which kinds of rational functions of one variable have an inverse relation that contains a branch that is a rational function?
Which kinds of polynomial functions of one variable have an inverse relation that contains a branch that is a rational function?

horrupavinvo 2022-02-15

I have
$\sum_{n = 0}^{\infty} {(- 1)}^{n} 2 n x^{(2 n - 1)}$
It turns out that this series is equal to the function $\frac{- 2 x}{{(1 + x^{2})}^{2}}$
Is there a general method that would demonstrate this fact beforehand?

The true proportion p of people who give a favorable rating to Congress is 8% with a margin of error of1.5%. Describe this statement using an absolute value equation.my answer key has the answer as 1 but that doesn't seem correct what am I doing wrong or are they wrong

Stefan Hendricks 2021-12-20

A manufacturer of lighting fixtures has daily production costs of $C = 800 - 10 x + 0.25 x^{2} .$ where C is the total cost (in dollars) and X is the number of units produced. How many fixtures should be produced each day to yield a minimum cost?

Perform the indicated operation h(n) = 4n-4 g(n) = n^2+3n find (h*g)(2)

jack89515lg 2021-12-01

True or False. Every rational function has at least one vertical asymptote.

Line 2021-10-26

The rate of return.
Given information:
$\begin{array}{cc} Y e a r & C a s h F l o w & I R R @ 39 % & P r e s e n t V a l u e \\ 0 & - $ 200 & 0 & - 200 \\ 1 & 350 & 0.719 & 251.65 \\ 2 & - 100 & 0.517 & - 51.70 \end{array}$
As PV at $39 %$ is zero. So, the rate of return is $39 %$ .
PV Value at $39 %$ is calculated using the table.

illusiia 2021-09-25

The process by which we determine limits of rational functions applies equally well to ratios containing noninteger or negative powers of x: Divide numerator and denominator by the highest power of x in the denominator and proceed from there. Find the limits. $lim_{x \to \infty} \frac{x - 3}{\sqrt 4 x^{2} + 25}$

texelaare 2021-09-25

Use substitution to convert the integrals to integrals of rational functions. Then use partial fractions to evaluate the integrals. $\int \frac{\cos x}{\sin x (1 - \sin x)} d x$

vazelinahS 2021-09-24

Determine
$lim_{x \to \infty} f (x)$
and
$lim_{x \to - \infty} f (x)$
for the following rational functions. Then give the horizontal asymptote of f (if any).
$f (x) = \frac{- x^{3} + 1}{2 x + 8}$

Tahmid Knox 2021-09-24

For the following exercises, find the domain of the rational functions. $f (x) = \frac{x^{2} + 4}{x^{2} - 2 x - 8}$

Sinead Mcgee 2021-09-22

(a) rewrite the sum as a rational function S(n), (b) use S(n) to complete the table, and (c) find $lim_{n \to \infty} S (n)$
$S (n) = 100, 101, 102, 103, 104$
$\sum_{i = 1}^{n} \frac{3}{n^{3}} (1 + i^{2})$

Tazmin Horton 2021-09-21

Use substitution to convert the integrals to integrals of rational functions. Then use partial fractions to evaluate the integrals.
$\int \frac{1 + ϵ^{x}}{1 - ϵ^{x}} d x$

Cheyanne Leigh 2021-09-19

Use substitution to convert the integrals to integrals of rational functions. Then use partial fractions to evaluate the integrals. $\int \frac{d x}{\sqrt{x} + \sqrt[4]{x}}$

Rational functions equations are used daily to solve rational issues that can help with the approximation and calculation of certain behavior. For example, rational functions solving is used in the weather forecast and when trying to find rationality. The answers to the questions that have been provided will act as your personal help as you approach answers as your rational functions solver online. Of course, you may compare what you have with the presented solutions. Make sure to look into rational function questions and answers and study the initial instructions, too.

Algebra II

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