Do regard a crossed horizontal asymptote while finding range , or not since it was crossed

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If a horizontal asymptote is crossed after reciprocating an exponential function , should I include the horizontal asymptote when trying to find the range of my new function .

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Your wording a a little confusing. A picture would help. But I think I can guess what you mean.

The answer to your question is probably "yes". If there is an $x$ such that $f(x) = h$ then $h$ is in the range of $f$ whether or not somewhere else it's the height of a horizontal asymptote.