Inverse Rational Functions

Suppose  y = f(x) = P(x) / Q(x)  is a rational function, with  P  and  Q  polynomials. If we rewrite this as  y Q(x) – P(x) = 0 , we see that solving for  x  in terms of  y  amounts to finding a root of a polynomial equation. Previous observations about finding polynomial inverses, and polynomial roots, apply here as well.

There are many ways for a rational function to fail to have an inverse.

Multiple singularities, or singularities with multiplicity > 1, will not do:

Neither will eliminating singularities altogether:

Short of a denominator of  1 (which reduces the rational function to a polynomial), the only way to arrange for an invertible rational function is to have a horizontal asymptote and just one singularity:

In this case, we solve for  x  in terms of  y  as follows: