롤의 정리와 평균값 정리

If $2x+\sin x$ is one–to–one over an interval, it never takes the same value for two distinct numbers in that interval. By Rolle's Theorem, if $f(a)=f(b)$ for distinct $a$ and $b$, then $f'(c)=0$ for some $c$ between $a$ and $b$. However, $f'(x)=2+\cos x$, which is never zero. In fact, $f'(x)\ge 1$ for all $x$, so $f(x)$ is strictly increasing over its entire domain. Therefore, our function $f$ never takes the same value twice, so it is one–to–one over all the real numbers, $(-\infty,\infty)$.

When we define the inverse function $f^{-1}(x)$, the domain of $f$ is the range of $f^{-1}$, and vice-versa. In general, we might have to restrict the domain of $f$ (and hence the range of $f^{-1}$) to an interval where $f$ is one–to–one, but in our case, this isn't necessary. So, the range of $f^{-1}$ is $(-\infty,\infty)$ and the domain of $f^{-1}$ is the range of $f$: $(-\infty,\infty)$.