Advance Polynomial & Rational Functions


1. A rational function has the form 𝑓(𝑥) = 𝑎𝑥^2/𝑥^𝑛−5 where a is a non-zero real number and n is a positive integer.

a) Are there any values of "a" and n for which the function above will have no vertical asymptotes? Explain why by providing examples and showing work. Use terminology learned in this unit. 

b) Are there any values of "a" and "n" for which the function above will have a horizontal asymptote? Explain why by providing examples and showing work. Use terminology learned in this unit. 
Functions
Mina Mina
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  • Paul F Paul F
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    This is a time consuming problem, and it may take about 20-25 minute to write a good solution. This type of questions should come with a bounty.

    • Mina Mina
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      me no have money 😞😞

    • Paul F Paul F
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      No worries, I will answer it as a Pro bono question.

    • Paul F Paul F
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1 Answer

a) For the function $𝑓(𝑥) =\frac{𝑎𝑥^2}{𝑥^𝑛}−5=a x^{2-n}-5$ to have no vertical asymptote, you must not be deviding by zero. This means that the exponent of $x$ must not be a negative number, i.e. 
\[2-n \geq 0.\]
Hence this function does not have a veritical asymptote if $n\leq 2$. Also of $a=0$, then $f(x)=-5$ and it does not have a veritical asymptote. In summary we should have
\[a=0   \text{or}  n\leq 2\]
to not have any vertical asymptote. For example if we let $n=0$, $a=2$. Then
\[f(x)=x^2-5,\]
which does not have any veritical asymptote. 

b) For this function $f(x)=a x^{2-n}-5$ to have a horizontal asymptote, we must have
\[2-n \leq 0         \text{or}      n\geq 2.\]
If $a=0$ then $f(x)=-5$, which has the veritical asymptote $y=-5$. So to have a horizontal asymptote we must have
\[a=0   \text{or}   n\geq 2.\]

Paul F Paul F
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