I need help solving the following function for $x$
$$g(x) = x - x \cdot \cosh\left(\frac{1}{2x}\right)$$
As I have never used hyperbolic functions, all my attempts at solving this have failed miserably. The closest that I got was (using hightschool math)
$$\frac{1}{2x} = \cosh^{-1}\left(1 - \frac{g}{x}\right)$$
but that is going nowhere.
Note: a similar question has been asked here but it did not help me.
Hint
You cannot solve for $x$ the equation $$g(x) = x - x \, \cosh\left(\frac{1}{2x}\right)$$ because it is a transcendental equation.
But what you could do is to expand the rhs as a series for large values of $x$ (better will be to let $x=\frac 1t$) and you will get $$g(x)=\sum_{n=0}^\infty \frac {a_n}{x^n}$$ Take very few terms and use series reversion to have an approximation of $x$ as a function of $g(x)$.
You will (may be) be surprised to see how good it is.