Determine the linear approximation of $g$ about $2\pi$ given that $g$ is the inverse of $f$

Question

For all real numbers of $x$ $f(x)=2x+sin(x)$ Show that $f$ has an inverse function $g$ and determine the domain of $g$. Determine the linear approximation of $g$ about $2\pi$.

$f$ Has an inverse function, because the derivative of $f$ is $f'$=$2+cos(x)$, so $f$ is one-to-one and thus has an inverse. The domain of $g$ is the range of $f$, which is $(-\infty, \infty)$.

I don't know how i can find the linear approximation of $g$ about $2\pi$.

Answer 1

What you did is alright, and now use the theorem for the derivative of the inverse function:

$$y=f(x)=2x+\sin x\implies g(x)=f^{-1}(x),\;\text{and its linear approximation about $2\pi$ is}$$

$$ g(x)=L(2\pi)=g(2\pi)+g'(2\pi)(x-2\pi) $$

observe that $\;f(\pi)=2\pi\;$ , so $\;f^{-1}(2\pi)=\pi\;$ and thus

$$g'(2\pi)=\frac1{f'(f^{-1}(2\pi))}=\frac1{f'(\pi)}=\frac1{2\pi+\cos\pi}=\frac1{2\pi-1}\implies$$

$$L(2\pi)=\pi+\frac{x-2\pi}{2\pi-1}$$