Expert: Sherman D. - 6/2/2005

Question
The graph of y=x/(1-absolute value(x)) has
A. no horizontal asymptotes and one vertical asymptote
B. one horizontal asymptote and one vertical asymptote
C. two horizontal asymptotes and one vertical asymptote
D. one horizontal asymptote and two vertical asymptotes
E. two horizontal asymptotes and two vertical asymptotes
From my graphing calculator I can see the answer is E,but how would you know that without a graphing calculator? How do you do it algebraically?

Answer
y = x/(1 - |x|)

lets do it this way

whatever value makes the denominator 0 will give you vertical asymptotes

1 - |x| = 0

-|x| = -1

|x| = 1

as we know, if "x" is negative or positive, with the absolute value, you will still get a positive value, so

|x| = 1

think of this like this

±(x) = 1
x = ±1

and that is why you get 2 asymptotes

that gives you the 2 vertical asymptote part.

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y = x/(1 - |x|)
x = y/(1 - |y|)
x(1 - |y|) = y
x - |y|x = y
x = y + |y|x
Or you can say
x = y ± (y)x
x = y(1 ± x)
y = (x/(1 ± x))
this will give you 2 horizontal asymptotes at ±1

So that is how you get 2 sets of asymptotes.