How do you graph $f\left(x\right)=\frac{{x}^{2}-1}{x}$ using holes, vertical and horizontal asymptotes, x and y intercepts?

tonan6e
2022-09-30
Answered

How do you graph $f\left(x\right)=\frac{{x}^{2}-1}{x}$ using holes, vertical and horizontal asymptotes, x and y intercepts?

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Maddox Koch

Answered 2022-10-01
Author has **7** answers

You know this graph can't exist at x=0, since that would make the denominator equal 0. Because the polynomial on the top is of a bigger degree, there is a slant asymptote. Dividing the initial terms, we get $\frac{{x}^{2}}{x}=x$, so there is a slant asymptote at y=x.

Since we can factor the top into (x−1)(x+1), we know the function has two solutions at $x=\pm 1$.

Plotting these solutions and following the asymptotes makes this a straightforward graph to sketch: graph{(x^2-1)/x [-10, 10, -5, 5]}

Also, not all rational functions are so easy to pblackict the behavior of, so creating a table of x and y values is always a good idea! And if you need more information about how to find the asymptotes, look here.

Since we can factor the top into (x−1)(x+1), we know the function has two solutions at $x=\pm 1$.

Plotting these solutions and following the asymptotes makes this a straightforward graph to sketch: graph{(x^2-1)/x [-10, 10, -5, 5]}

Also, not all rational functions are so easy to pblackict the behavior of, so creating a table of x and y values is always a good idea! And if you need more information about how to find the asymptotes, look here.

asked 2021-02-25

True or False. The graph of a rational operate could encounter a horizontal straight line.

asked 2022-06-20

For a field $k$, let $V$ be an affine variety over $k$. Denote by $k(V)$ the function field of $V$, containing all rational functions $r:V\u21e2{\mathbb{A}}_{k}^{1}$. My question is, if a rational function $f\in k(V)$ has a pole at $p\in V$, is there an expression $f=\frac{g}{h}$ where $g,h\in k[V]$ are regular functions, and $g(p)\ne 0$, $h(p)=0$?

When $V\subseteq {\mathbb{A}}_{k}^{1}$, this is clear, since if we have $f=\frac{g}{h}$ where $g(p)=h(p)=0$, we can simply reduce the expression of $g$ and $h$ and eliminate the factor $(x-p)$ until we get $f=\frac{{g}^{\prime}}{{h}^{\prime}}$ such that ${g}^{\prime}(p)\ne 0$, ${h}^{\prime}(p)=0$. But when $g,h$ are multivariate functions, I wonder how to get such a reduced expression?

When $V\subseteq {\mathbb{A}}_{k}^{1}$, this is clear, since if we have $f=\frac{g}{h}$ where $g(p)=h(p)=0$, we can simply reduce the expression of $g$ and $h$ and eliminate the factor $(x-p)$ until we get $f=\frac{{g}^{\prime}}{{h}^{\prime}}$ such that ${g}^{\prime}(p)\ne 0$, ${h}^{\prime}(p)=0$. But when $g,h$ are multivariate functions, I wonder how to get such a reduced expression?

asked 2021-09-11

Determine

$\underset{x\to \mathrm{\infty}}{lim}f\left(x\right)$

and

$\underset{x\to -\mathrm{\infty}}{lim}f\left(x\right)$

for the following rational functions. Then give the horizontal asymptote of f (if any).

$f\left(x\right)=\frac{4{x}^{2}-7}{8{x}^{2}+5x+2}$

and

for the following rational functions. Then give the horizontal asymptote of f (if any).

asked 2022-09-02

If f(x) varies directly with x and f(x) = 24 when x = –4, then what is f(x) when x = 12?

asked 2022-06-24

I want to decompose the rational function

$\frac{{\textstyle P(s)}}{{\textstyle Q(s)}}=\frac{{\textstyle \prod _{i=1}^{m}(s+{a}_{i})}}{{\textstyle \prod _{i=1}^{n}(s+{b}_{i})}}$

where ${a}_{i}>0$ for every $i=1,\dots ,m$ ${b}_{i}>0$ for every $i=1,\dots ,n$ and $n>m$.

In other words, I'm looking for coefficients ${x}_{j}$, $j=1\dots ,n$ such that

$\frac{{\textstyle P(s)}}{{\textstyle Q(s)}}=\frac{{\textstyle {x}_{1}}}{{\textstyle s+{b}_{1}}}+\cdots +\frac{{\textstyle {x}_{n}}}{{\textstyle s+{b}_{n}}}\phantom{\rule{1em}{0ex}}\u27fa\phantom{\rule{1em}{0ex}}\sum _{j=1}^{n}{x}_{j}\prod _{{\scriptstyle \begin{array}{c}i=1\\ i\ne j\end{array}}}^{n}(s+{b}_{i})=\prod _{i=1}^{m}(s+{a}_{i})$

Making some attempts with Mathematica for low values of $m$ and $n$ it seems that the ${x}_{j}$ are given by

${x}_{j}=\frac{{\textstyle \prod _{i=1}^{m}({a}_{i}-{b}_{j})}}{{\textstyle \prod _{{\scriptstyle \begin{array}{c}i=1\\ i\ne j\end{array}}}^{n}({b}_{i}-{b}_{j})}}$

So my questions are:

1) Can I say beforehand that there exist unique such ${x}_{j}$s?

2) How can I prove that ${x}_{j}$ has in general (as it seems) the form above?

$\frac{{\textstyle P(s)}}{{\textstyle Q(s)}}=\frac{{\textstyle \prod _{i=1}^{m}(s+{a}_{i})}}{{\textstyle \prod _{i=1}^{n}(s+{b}_{i})}}$

where ${a}_{i}>0$ for every $i=1,\dots ,m$ ${b}_{i}>0$ for every $i=1,\dots ,n$ and $n>m$.

In other words, I'm looking for coefficients ${x}_{j}$, $j=1\dots ,n$ such that

$\frac{{\textstyle P(s)}}{{\textstyle Q(s)}}=\frac{{\textstyle {x}_{1}}}{{\textstyle s+{b}_{1}}}+\cdots +\frac{{\textstyle {x}_{n}}}{{\textstyle s+{b}_{n}}}\phantom{\rule{1em}{0ex}}\u27fa\phantom{\rule{1em}{0ex}}\sum _{j=1}^{n}{x}_{j}\prod _{{\scriptstyle \begin{array}{c}i=1\\ i\ne j\end{array}}}^{n}(s+{b}_{i})=\prod _{i=1}^{m}(s+{a}_{i})$

Making some attempts with Mathematica for low values of $m$ and $n$ it seems that the ${x}_{j}$ are given by

${x}_{j}=\frac{{\textstyle \prod _{i=1}^{m}({a}_{i}-{b}_{j})}}{{\textstyle \prod _{{\scriptstyle \begin{array}{c}i=1\\ i\ne j\end{array}}}^{n}({b}_{i}-{b}_{j})}}$

So my questions are:

1) Can I say beforehand that there exist unique such ${x}_{j}$s?

2) How can I prove that ${x}_{j}$ has in general (as it seems) the form above?

asked 2022-02-18

After having read abstract concepts of algebraic curves, I have trouble dealing with actual examples. For instance, why is the $\varphi =\frac{y}{x}$ a rational function on the curve $F={y}^{2}+y+{x}^{2}$ ? I know that any rational function on this curve should be of the form $\{\varphi =\frac{f}{g}:f,g\in \frac{K[x,y]}{\left(F\right)},g\ne 0\}$ , but what do I need to actually check to show that this is a rational function on F?
Any help will be good

asked 2022-09-02

If y varies inversely as the cube of x and directly as the square of z and y = -6 when x=3 and z =9, how do you find y when x =6 and z= -4?