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Find the range of the function $$f(x) = \frac{1}{1-x} + \frac{1}{1 + x}.$$ Express your answer in interval notation.

 Aug 12, 2023
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f(x) =   1 / (1 - x)  + 1/ ( 1 + x)      rewrite as

 

f(x) =  2 / ( 1 - x^2)

 

Clearly, x cannot be either -1 or 1 because these two  values make the  denominator = 0

 

We have  a lower power polynomial  divided by a higher power polynomial....in such a case we will have a horizontal asymptote at y = 0  and it will approach -inf   as x approaches  -1  from the  left and  1 from the right

 

So...part of the range is (-inf, 0)

 

And the function  will have a minimum positive  value of 2 when x = 0 and  it will verge towards infinity whenever x approaches 1 from the left or -1 from the  right

 

So...the whole range is   (-inf,0) U [2, inf)

 

cool cool cool

 Aug 13, 2023

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