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Does the function $$y(x) = cx + a \sinh (bx) $$ have an inverse function $x=x(y)$ and if yes, what is it?

The part $$y(x) = a \sinh(bx)$$ can easily be inverted to $$ x(y) = asinh(y/a)/b$$

For $c>0$ the function is strictly monotonic, so by just looking at the graph I would assume an inverse should exist.

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    If $c=-10$ and $a=b=1$, then the inverse is not definded for $x \in \mathbb{R}$. So for arbitrary values of $a,b,c$ we can reject that the inverse does exist for all $x\in \mathbb{R}$. But if you choose specific values of $a,b,c$ the inverse does exist for all $x \in \mathbb{R}$. But I highly doubt that you can state the inverse for non trivial choices of $a,b,c$ as a closed form solution. – 2017-01-20
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    Thanks, I have now updated the question, a solution for positive c is what I was actualy looking for. – 2017-01-20
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    As I said it is very unlikely that you will be able to get a closed form solution for the inverse. What do you plan to do with the inverse? – 2017-01-20
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    Use both functions in a simulation tool, which is capable of simplifying the system of equations if inverses and or derivatives are given. A discussion is here: https://github.com/iea-annex60/modelica-annex60/issues/635#issuecomment-273116120, I have also asked two more related questions in the last days. – 2017-01-20
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    related: http://math.stackexchange.com/questions/2100016/solve-system-of-two-trigonometric-equations – 2017-01-20
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    related: http://math.stackexchange.com/questions/2096077/invert-regularisation-function – 2017-01-20
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    Would a taylor series of the inverse function help you? – 2017-01-20
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    The function and its inverse just have to be consistent numerically. – 2017-01-20
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    Let us [continue this discussion in chat](http://chat.stackexchange.com/rooms/52136/discussion-between-mryoumath-and-matth). – 2017-01-20

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Rule of thumb: if the unknown is both outside and inside trig/log/exp functions, there is no closed form inverse (unless in terms of purpose-defined special functions).

So no... just looking at your expression, seeing $x$ in a linear term and inside $\sinh$, the only way to do this is solving it numerically.