Why can't the inverse of $F(x)= x+\sin(x)$ have a formula algebraically?

Question

I'm only curious why the inverse of $f(x)$ can not be determined algebraically. Is it because the inverse of $\sin(x)$ cannot be converted into a formula?

Answer 1

Note that the function $f(x) = x + \sin x$ is transcendental function. Also we have a theorem which says that inverse of algebraic functions are also algebraic, therefore it follows that the inverse of $f$ is not algebraic.

That said I think you want to know something more. The function $f$ is transcendental but elementary (meaning that value of $f(x)$ can be obtained from $x$ via a finite number of algebraic operations together with composition of algebraic functions, trigonometric functions (direct and inverse), exponential and logarithmic functions). The inverse function is also transcendental but not elementary (it is difficult to prove in general that a function is non-elementary). Hence it is not possible to expect inverse of $f$ to be expressed in terms of usual functions we see in analysis.

Answer 2

The equation $x + \sin(x) = y$ is Kepler's equation in the case $e = -1$. This has been well-studied. It just happens that the inverse function is not one of the special functions to which standard names have been attached.