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# What are the asymptotes? And more...

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Function:

y = 1 / ((x+1)(x-1))

Find the asymptotes.

Specify what values y can be. (How to find the range of the function?)

Help is appreciated. I already know the answer from desmos calculator but I'd like to know how to find the asymptotes and find the range of the function without desmos.

Thanks =)

Oct 24, 2018

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y  =    1  / [ (x + 1) (x - 1) ]  =   1 / ( x^2  - 1 )

The  vertical asymptotes  are the x values that make the denominator  = 0

These are the lines x  = -1   and x  = 1

Since we have a lower degree polynomial over a higher degree polynomial...we will have a horizontal asymptote of 0   [ this will always be true in a  lower / higher   situation ]

The range is a little difficult

Note that when x is 0, the function will reach its maximum negative value  of -1

When  x  is either a large negative  or positive value   1 / [ x^2  - 1 ]  will be almost  0

As x  moves  closer  to the left   vertical  asymptote  x  = -1  from the left side  or  closer to the right vertical asymptote x = 1 from the right side...the function aprroaches infinity

For example.....let x  =   -1.0001   or   1.0001  and we get the large positive value  of ≈ 5000

And this will increase as we get closer to each of these asymptotes in this manner

As the function moves closer to the left asymptote of x  = -1 form the right or closer to  the right vertical asymptote of x = 1 from the left, the function approaches negative infinity

For example when  x  =  -.9999  or   x  =.9999 the function  value   ≈ -5000

And this will decrease as we get closer to these aymptotes in this manner

So....the range is  ( -infinity , 1 ] U  (0 , infinity )

Oct 24, 2018
#2
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Did you use any graphic calculator to find the answers or did you figure it out any other way? :)

Oct 24, 2018
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I DID look at the graph to make sure that I gave you the correct answers......but...it is also helpful to think about WHY  the graph looks as it does.....

CPhill  Oct 24, 2018