If I were to find an interation function for
$$4\sin(x)-e^x$$
I would do something like:
$$e^x = 4\sin(x)\implies x = \ln 4\sin(x) \implies x_{n+1} = \ln 4\sin(x_n)$$
or something with $\arcsin$, but I don't know how to arrive at $$\phi(x) = x -2\sin(x)+\frac{1}{2}e^x$$
But if I try to revserse it:
$$x= x -2\sin(x)+\frac{1}{2}e^x \implies 2\sin(x) = \frac{1}{2}e^x \implies 4\sin(x) = e^x \implies 4\sin(x)-e^x = 0$$
Can I just add $x$ and subtract like this? How it helps me in the iteration process?