How to find inverse of the function $f(x)=\sin(x)\ln(x)$ My friend asked me to solve it, but I can't.
If $f(x)=\sin(x)\ln(x)$, what is $f^{-1}(x)$?
I have no idea how to find the solution. I try to find
$$\frac{dx}{dy}=\frac{1}{\frac{\sin(x)}{x}+\ln(x)\cos(x)}$$ 
and try to solve it for $x$ by some replacing and other things, but I failed.
Can anyone help? Thanks to all.

In fact it has an infinity of solutions for any real value of $y=f(x)$ as may be seen on this picture by Alpha (nothing special happens near $0$ because of the equivalence to $x\ln(x)$) :

You may find solutions numerically by iterations.
For example using Newton-Raphson applied to $\log(x)\sin(x)=y$ we get the first solution illustrated (at the left) in the case $y=1$ :
$$x_{n+1}=x_n-\frac{\sin(x_n)\log(x_n)-y}{\cos(x_n)\log(x_n)+\sin(x_n)/x_n},\quad x_0=7$$
$$s_1\doteq 6.83056530451751861265\cdots$$
