We know that for rational functions f(x)/g(x) vertical asymptotes are defined as the lines x=x0 where g(x0)=0, and for horizontal asymptotes it is the limit as x approaches infinity or x approaches −infinity. 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),

Marcelo Mullins

Marcelo Mullins

Answered question

2022-07-17

Vertical/horizontal asymptotes
We 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.

Answer & Explanation

phinny5608tt

phinny5608tt

Beginner2022-07-18Added 17 answers

Good question, and the answer is not a simple "yes" or "no".
First of all, it's not quite true what you said about vertical asymptotes. If we'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=0
Now, when inverting a function y=f(x), we need to be sure that the function f is one-to-one. If it'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'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'm assuming here that we'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're going to have a hard time finding an interval on which to invert that function and see a horizontal asymptote.
Awainaideannagi

Awainaideannagi

Beginner2022-07-19Added 5 answers

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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