How can I find an output of this function's inverse without graphing?

Question

How can I find $f^{-1}(5)$ where $$f(x)=\frac{27}{\pi}x + \sin x$$ algebraically? Thank you!!

Answer 1

From $$ f(x)=\frac{27}{\pi}x + \sin x,\quad x \in \mathbb{R}, $$ we get $$ f'(x)=\frac{27}\pi+\cos x>0,\quad x \in \mathbb{R}, $$ the function $f$ is then strictly increasing over $\mathbb{R}$, observing that $$ f\left(\frac\pi6\right)=\frac{27}\pi\times\frac\pi6+\frac12=\frac{10}2=5 $$ gives

$$ f^{-1}(5)=\frac\pi6 $$

(without graphing).