Noa I.

asked • 06/14/15

Write a rational function (f) that has the specified characteristics. vertical asymptote: x=-2, slant asymptote: y=x+1, zero: x=2

 I've tried everything but only find that the answer is not possible.
There may be a hole.
 
Write a rational function (f) that has the specified characteristics.
vertical asymptote: x=-2
slant asymptote: y=x+1
zero: x=2

1 Expert Answer

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Mark M. answered • 06/14/15

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vertical asymptote, the denominator must contain (x + 2)
zero the numerator must contain (x - 2)
 
So far f(x) = (x - 2) / (x + 2)
 
For slant asymptote, quotient of numerator divided by denominator must be (x + 1)
 
Therefore:
 
f(x) = (x + 1)(x - 2) / (x + 2)
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Noa I.

I got the same answer however I found it to be impossible because for there to be a slant asymptote, the zero needs to be crossed out.
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06/14/15

Mark M.

What do you mean by "zero needs to be crossed out?"
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06/14/15

Noa I.

Even if "zero needs to be crossed out" doesn't make sense, the function you found wouldn't work when finding the slant asymptote. The slant asymptote would end up being y=x+3 with your function.
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06/14/15

Mark M.

Starting from f(x) = (x - 2) / (x + 2)
 
Another polynomial, call it P, must be determined such that
 
P(x - 2) / (x + 2) = x + 1 + some remainder
P(x - 2) = x2 + 3x + 2 + some remainder
P = (x2 + 3x + 2) / (x - 2)
P = x + 5
 
f(x) = (x + 5)(x - 2) / (x + 2)
f(x) = (x2 + 3x - 10) / (x + 2)
 
(x2 + 3x - 10) / (x + 2) = x + 1 - 12/(x+2)
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06/14/15

Noa I.

Is (x2 + 3x - 10) / (x + 2) = x + 1 - 12/(x+2) your final answer to the original question?
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06/14/15

Mark M.

Reminds me of a quiz show.
f(x) = (x2 + 3x - 10) / (x + 2)
Final answer.
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06/14/15

Noa I.

How do I explain how I got x+5 when I factor out the numerator?
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06/14/15

Mark M.

x2 + 3x - 10 = (x + 5)(x - 2)  byFOIL
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06/14/15

Noa I.

No I get that I can get it by factoring, but 5 isn't a number in the specified characteristics so it comes out as a random number.
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06/14/15

Mark M.

Read my solution above  starting with: Starting from f(x) = (x - 2) / (x + 2).
There is where the factor (x + 5) is derived.
Tell me what I have not made clear.
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06/14/15

Noa I.

I see it now. Thank you so much! 
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06/14/15

Mark M.

You are welcome so much!
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06/14/15

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