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Question 89509: Write a rational function satisfying the following criteria:
Vertical assymptote: x=1
Assymptote: y=x+1
zero of the function: x=-2
Answer by stanbon(75887)   (Show Source):
You can put this solution on YOUR website! Write a rational function satisfying the following criteria:
Vertical asymptote: x=1
so f(x) must have a factor of (x-1) in the denominator
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Asymptote: y=x+1
So, f(x) = g(x)/(x-1) = x+1 + k/(x-1)
Need to find g(x) and k
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zero of the function: x=-2
so f(x) must have a factor of (x+2) in the numerator.
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Working the problem:
f(x) = [(x+2)(ax+b)/(x-1)] = x+1 + k/(x-1)
Then:
f(-2) = 0 = -2+1+k/(-2-1)
k/-3 = 1
k = -3
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f(x) = [(ax^2+(b+2a)x+2b)/(x-1)] = x+1 -3/(x-1)
Need to find a and b:
Divide ax^2+(b+2a)x+2b by (x-1) to force a quotient of (x+1) and remainder =-3:
This will happen if a=1 and b = -2
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So, f(x)= [(x+2)(x-2)/(x-1)]
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Checking:
(x^2-4)/(x-1) = x+1 -3/(x-1) which has a vertical
asymptote at x=1, a zero at x=-2, and an asymptote of x+1
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Cheers,
Stan H.
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