Why isn't $y=\frac{\sin x }{\pi}+x$ invertible

Question

Why isn't $\displaystyle y=\frac{\sin x }{\pi}+x$ invertible, i.e. solvable for $x$?

Each $x$ has a unique $y$ so it should be invertible.

http://www.wolframalpha.com/input/?i=solve+y%3Dsin(x)%2FPi%2Bx+wrt+x

Answer 1

Each $x$ has a unique $y$ holds for any function $y=f(x)$, but obviously not every function is invertible. Continuous functions are invertible if monotonic, and $$ f(x)=\frac{\sin x}{\pi}+x $$ is an increasing function since $$ f'(x)= \frac{\cos x}{\pi}+1 > 0.$$