equipokypip1
2022-09-13
Answered

How do you graph $f\left(x\right)=\frac{8}{x(x+2)}$ using holes, vertical and horizontal asymptotes, x and y intercepts?

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Raven Mosley

Answered 2022-09-14
Author has **14** answers

holes: a value that causes both the numerator and denominator to equal zero. there are no holes in this rational function.

vertical asymptotes: it's a line $\to$ set the denominator of the rational function equal to 0:

x(x+2)=0

vertical asymptotes: x=0,x=−2

horizontal asymptotes:

the following are the rules for solving horizontal asymptotes:

let m be the degree of the numerator

let n be the degree of the denominator

if m > n, then there is no horizontal asymptote

if m = n, then the horizontal asymptote is dividing the coefficients of the numerator and denominator

if m < n, then the horizontal asymptote is y=0.

As we can see in our rational function, the denominator has a larger degree of x. So the horizontal asymptote is y=0.

x-ints: x-intercepts are the top of the rational function. Since the numerator just says 8, that means that there are no x-ints.

y-ints: y-intercepts are when you plug in 0 to the function:

$\frac{8}{0(0+2)}$

$\frac{8}{0}\to$ undefined, so there are no y-ints.

vertical asymptotes: it's a line $\to$ set the denominator of the rational function equal to 0:

x(x+2)=0

vertical asymptotes: x=0,x=−2

horizontal asymptotes:

the following are the rules for solving horizontal asymptotes:

let m be the degree of the numerator

let n be the degree of the denominator

if m > n, then there is no horizontal asymptote

if m = n, then the horizontal asymptote is dividing the coefficients of the numerator and denominator

if m < n, then the horizontal asymptote is y=0.

As we can see in our rational function, the denominator has a larger degree of x. So the horizontal asymptote is y=0.

x-ints: x-intercepts are the top of the rational function. Since the numerator just says 8, that means that there are no x-ints.

y-ints: y-intercepts are when you plug in 0 to the function:

$\frac{8}{0(0+2)}$

$\frac{8}{0}\to$ undefined, so there are no y-ints.

asked 2021-02-25

True or False. The graph of a rational function may intersect a horizontal asymptote.

asked 2022-05-23

Let

$f(z)={\displaystyle \frac{z-a}{z-b}},\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}z\ne b\ne a$

be a complex valued rational function.

How can I show that, if $|a|,|b|<1,$, then there is a complex number ${z}_{0}$ satisfying $|{z}_{0}|=1$ and $f({z}_{0})\in \mathbb{R}$ ?

I have tried in many ways, but on success. Basically I tried to show that there is a unimodular complex number such that

$\frac{a-b}{z-b}}={\displaystyle \frac{\overline{a}-\overline{b}}{\overline{z}-\overline{b}}}.$

I could make a quadratic equation by using the fact that $\overline{z}={\displaystyle \frac{1}{z}},\phantom{\rule{thinmathspace}{0ex}}\mathrm{\forall}z\in \mathrm{\partial}\mathbb{D}.$ Unfortunately I could not solve this question using that. So, I would like to see different (and somewhat general) approach.

Also, I would like to know that what happen if (both or atleast one) $a,b\notin \mathbb{D}.$Any comment or hint will be welcome. Thank you.

$f(z)={\displaystyle \frac{z-a}{z-b}},\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}z\ne b\ne a$

be a complex valued rational function.

How can I show that, if $|a|,|b|<1,$, then there is a complex number ${z}_{0}$ satisfying $|{z}_{0}|=1$ and $f({z}_{0})\in \mathbb{R}$ ?

I have tried in many ways, but on success. Basically I tried to show that there is a unimodular complex number such that

$\frac{a-b}{z-b}}={\displaystyle \frac{\overline{a}-\overline{b}}{\overline{z}-\overline{b}}}.$

I could make a quadratic equation by using the fact that $\overline{z}={\displaystyle \frac{1}{z}},\phantom{\rule{thinmathspace}{0ex}}\mathrm{\forall}z\in \mathrm{\partial}\mathbb{D}.$ Unfortunately I could not solve this question using that. So, I would like to see different (and somewhat general) approach.

Also, I would like to know that what happen if (both or atleast one) $a,b\notin \mathbb{D}.$Any comment or hint will be welcome. Thank you.

asked 2021-09-17

Determine

and

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

asked 2022-07-02

I have developed the formula to determine the radius of a cylinder with a fixed volume:

$f(x)=\sqrt[3]{{\displaystyle \frac{V}{\pi}}}\text{}$

Substituted into the formula for the surface area of a cylinder, I get the following function. This would give me the minimum surface area of a cylinder for a given volume.

$S(V)=2\pi (\sqrt[3]{{\displaystyle \frac{V}{\pi}}}{)}^{2}+2\pi (2\ast \sqrt[3]{{\displaystyle \frac{V}{\pi}}})$

However, my assignment for class asks for a rational function for this problem. How could I take my existing function and make it rational?

$f(x)=\sqrt[3]{{\displaystyle \frac{V}{\pi}}}\text{}$

Substituted into the formula for the surface area of a cylinder, I get the following function. This would give me the minimum surface area of a cylinder for a given volume.

$S(V)=2\pi (\sqrt[3]{{\displaystyle \frac{V}{\pi}}}{)}^{2}+2\pi (2\ast \sqrt[3]{{\displaystyle \frac{V}{\pi}}})$

However, my assignment for class asks for a rational function for this problem. How could I take my existing function and make it rational?

asked 2022-04-10

Compute all the points of discontinuity of $\frac{{a}^{2}-25a+154}{{a}^{5}-10{a}^{4}+21{a}^{3}+28{a}^{2}-52a-48}$.

asked 2022-05-24

If I have two algebraic numbers $\alpha $ and $\beta $ and a rational function $w$ with rational coefficients (a function that's the ratio of two rational polynomials) that relates the two $\alpha =w(\beta )$ if I were to substitute $\beta $ with one of it's algebraic cojugates ${\beta}^{{}^{\prime}}$ in the rational function will I get a conjugate of $\alpha $ or rather is $w({\beta}^{{}^{\prime}})$ an algebraic conjugate of alpha? I feel like there is a very obvious counter example but I've been struggling to find one.

asked 2022-09-01

If y varies inversely as x and y = -3 when x = 6, how do you find x when y is 36?