Vertical/horizontal asymptotesWe know that for rational functions f ( x ) / g ( x )vertical asymptotes are defined as the lines x = x 0 where g ( x 0 ) = 0, and for horizontal asymptotes it is the limit as x approaches ∞ or x approaches − ∞. My question is the following: what is the relationship between vertical and horizontal asymptotes? For instance, if we have y = f ( x )and we express x as function of y, or x = f − 1 ( y ) ,is the horizontal asymptote of inverse function the same as the vertical one of the original?Thanks in advance.

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Vertical/horizontal asymptotesWe know that for rational functions  f  (  x  )      /    g  (  x  )vertical asymptotes are defined as the lines   x  =      x    0   where   g  (      x    0    )  =  0, and for horizontal asymptotes it is the limit as x approaches   ∞ or x approaches   −  ∞. My question is the following: what is the relationship between vertical and horizontal asymptotes? For instance, if we have  y  =  f  (  x  )and we express x as function of y, or  x  =      f          −      1        (  y  )  ,is the horizontal asymptote of inverse function the same as the vertical one of the original?Thanks in advance.

phinny5608tt · Accepted Answer

Good question, and the answer is not a simple &quot;yes&quot; or &quot;no&quot;.First of all, it&#039;s not quite true what you said about vertical asymptotes. If we&#039;re considering   y  =            f      (      x      )              g      (      x      )      , points where g(x)=0 could give rise to either vertical asymptotes or to removable discontinuities. For example,   y  =            sin      ⁡      x        x   does not have a vertical asymptote at x=0Now, when inverting a function y=f(x), we need to be sure that the function f is one-to-one. If it&#039;s not, we may be able to obtain various invertible functions by restricting the domain of f in different ways. For example: There is no inverse for the function   f  (  x  )  =      x    2   defined on all reals, but if we restrict its domain to   [  0  ,  ∞  ), we get the inverse       f          −      1        (  y  )  =      y  , while restricting the domain instead to   (  −  ∞  ,  0  ] affords the inverse       f          −      1        (  y  )  =  −      y  If f is a function with a vertical asymptote at x=a, and we&#039;ve got some interval with a on its boundary and on which f is one-to-one, then yes. That vertical asymptote will turn into a horizontal asymptote for       f          −      1      .I&#039;m assuming here that we&#039;re talking about simple types of asymptotes. Check out the function   f  (  x  )  =            sin      ⁡              1        x              x   for an example of how weird things might get. You&#039;re going to have a hard time finding an interval on which to invert that function and see a horizontal asymptote.

Awainaideannagi · Accepted Answer

is the horizontal asymptote of inverse function the same as the vertical one of the original?For functions of the form   f  (  x  )  =                    a        x        +        b                    c        x        +        d              ,  c  ≠  0, this statement is true. The inverse is       f          −      1        (  x  )  =                    d        x        −        b                    −        c        x        +        a

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