Does $f(x) = x - \tanh(x)$ have an inverse function that can be expressed in terms of elementary functions? I find this question relevant in my current study of the tractrix, namely because this expression appears in one parameterization of the curve. I’ve noticed that the plot of the Cartesian equation of the tractrix appears to be the graph of a function. As you may have guessed, I became interested in discovering this function, and led myself to this question.
If $x - \tanh(x)$ does indeed have an elementary inverse, I’m more interested in how to derive an expression for it than the expression itself. If it doesn’t, I’d like to know why.
 A: Not a solution in elementary functions
We can do this with series reversion.  As $x \to 0$ we have
$$
x - \tanh x = 
{\frac{1}{3}}{x}^{3}-{\frac{2}{15}}{x}^{5}+{\frac{17}{315}}{x}^{7}-{
\frac{62}{2835}}{x}^{9}+O \left( {x}^{11} \right) 
$$
Solving $y=x-\tanh x$, we get as $y \to 0$
$$
x = {3}^{1/3}\,{y}^{1/3}+\frac{2}{5}\,y+{\frac {9\cdot{3}^{2/3}}{175}}{y}^{5/3}+{
\frac {2\cdot{3}^{1/3}}{175}}{y}^{7/3}+O \left( {y}^{3} \right) 
$$
A: Inverse functions are sometimes transcendental in nature, cannot be expressed in terms of elementary functions.
Way to derive 3D coordinates of asymptotic lines on a pseudosphere for tractrix , in polar/cylindrical coordinates $(r,\theta,z$ ):
$$ \sin \psi = \sin \phi = r/a = \frac{r d\theta}{ds}$$
$$ dr/ds= \sin \phi  \cos \psi, dz/ds=\cos \phi \cos \psi $$
Integrating with initial conditions $ (z=0, r=a) $ you get:
$$ z/a= \theta - \tanh \theta,\, r/a= \operatorname{sech} \,\theta. $$
We have parametric $ r(\theta), z(\theta)$ and there would perhaps be no  need to find an inverse function.

We can find $z$ for any $\theta.  $ Explicitly solving for $z$ is not possible analytically but possible only numerically.

Image from an earlier post here.
