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

Thordiswl
2022-09-17
Answered

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

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

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How do you write an inverse variation equations given y=8 when x=1.55?

asked 2022-06-25

I was just thinking that as rational functions form an ordered field you could describe analogous version of the absolute value function, but we don't quite have a 'metric' - for example |1/x| < e for all e > 0 in R but 1/x =/= 0.

I was wondering if anybody got anywhere with this 'metric' and if there are any links to papers exploring actual metrics on the rational functions?

I was wondering if anybody got anywhere with this 'metric' and if there are any links to papers exploring actual metrics on the rational functions?

asked 2022-06-21

I am thinking if I could help with my current problem. Now I have a parameterized rational function $G(p,z)$, where $p\in {\mathbb{R}}^{n}$ denotes the coefficients (parameters) of the rational function, and $z$ denotes the indeterminate of the rational function which lies in complex domain.

I regard $G(p,z)$ as a mapping from ${\mathbb{R}}^{n}$ to $\mathbb{G}$, where $\mathbb{G}$ is a set of rational functions with indeterminate $z$. Then I define that a property holds on a metric space $(\mathbb{G},d)$ if it holds on an open dense subset of $\mathbb{G}$.

However, I am wondering what conditions I should put on a parameter set $\mathrm{\Theta}\subseteq {\mathbb{R}}^{n}$, such that $\{G(p,z)|p\in \mathrm{\Theta}\}$ becomes an open subset of $\mathbb{G}$. Is making $\mathrm{\Theta}$ an open dense subset of ${\mathbb{R}}^{n}$ sufficient?

I regard $G(p,z)$ as a mapping from ${\mathbb{R}}^{n}$ to $\mathbb{G}$, where $\mathbb{G}$ is a set of rational functions with indeterminate $z$. Then I define that a property holds on a metric space $(\mathbb{G},d)$ if it holds on an open dense subset of $\mathbb{G}$.

However, I am wondering what conditions I should put on a parameter set $\mathrm{\Theta}\subseteq {\mathbb{R}}^{n}$, such that $\{G(p,z)|p\in \mathrm{\Theta}\}$ becomes an open subset of $\mathbb{G}$. Is making $\mathrm{\Theta}$ an open dense subset of ${\mathbb{R}}^{n}$ sufficient?

asked 2022-09-03

Is z=xy an inverse variation?