How do you find the asymptote and graph $y=\frac{1}{x-5}-4$?

sincsenekdq
2022-09-16
Answered

How do you find the asymptote and graph $y=\frac{1}{x-5}-4$?

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asked 2022-05-15

Express the following rational function in continued-fraction form:

$\frac{4{x}^{2}+3x-7}{2{x}^{3}+{x}^{2}-x+5}$

The answer is :

$\begin{array}{}\text{(inline continued fraction)}& \frac{4}{2x-\frac{1}{2}}+\frac{\frac{23}{8}}{x-\frac{63}{92}}-\frac{\frac{406}{529}}{x+\frac{33}{23}}\end{array}$

which means

$\frac{{\textstyle 4}}{{\textstyle 2x-\frac{1}{2}+\frac{{\textstyle \frac{23}{8}}}{{\textstyle x-\frac{63}{92}-\frac{{\textstyle \frac{406}{529}}}{{\textstyle x+\frac{33}{23}}}}}}}$

$\frac{4{x}^{2}+3x-7}{2{x}^{3}+{x}^{2}-x+5}$

The answer is :

$\begin{array}{}\text{(inline continued fraction)}& \frac{4}{2x-\frac{1}{2}}+\frac{\frac{23}{8}}{x-\frac{63}{92}}-\frac{\frac{406}{529}}{x+\frac{33}{23}}\end{array}$

which means

$\frac{{\textstyle 4}}{{\textstyle 2x-\frac{1}{2}+\frac{{\textstyle \frac{23}{8}}}{{\textstyle x-\frac{63}{92}-\frac{{\textstyle \frac{406}{529}}}{{\textstyle x+\frac{33}{23}}}}}}}$

asked 2022-02-17

From the definition of rational function $f\left(x\right)=\frac{p\left(x\right)}{q\left(x\right)}$ , where P(x) and q(x) are polynomials and $q\left(x\right)\ne 0$

so for the function$f\left(x\right)=\frac{{x}^{-2}+3}{x-5}$ by the definition f(x) isnt rational since the numerator is not polynomial but by multiplying both numerator and denominator by $x}^{2$ we get

$f\left(x\right)=\frac{3{x}^{2}+1}{{x}^{3}-5{x}^{2}}$ which is rational

and can we say that both functions are equal at every point?

so for the function

and can we say that both functions are equal at every point?

asked 2022-09-17

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