1. ## Inverse of x+sinx

Determine the inverse of the function:
$\displaystyle f(x)=x+sinx$

2. Originally Posted by pankaj
Determine the inverse of the function:
$\displaystyle f(x)=x+sinx$
First you must restrict your domain so that your function is one to one, and I am not sure this can be found? What makes you think it can? I mean we know that this has an inverse, but what makes you think it can be expressed in elementary terms?

3. The function is both one-one as well as onto since
$\displaystyle f'(x)=1+cosx\geq 0$ for all real x.i.e. $\displaystyle f(x)$ is an incresing function and will never repeat its value
From the graph it appears as if the inverse is $\displaystyle f^{-1}(x)=x-sinx$ since it appears to be reflection of $\displaystyle f(x)$ with respect to the line $\displaystyle y=x$ as mirror.I am not sure about this though.
I am looking for verification

4. Originally Posted by pankaj
The function is both one-one as well as onto since
$\displaystyle f'(x)=1+cosx\geq 0$ for all real x.i.e. $\displaystyle f(x)$ is an incresing function and will never repeat its value
From the graph it appears as if the inverse is $\displaystyle f^{-1}(x)=x-sinx$ since it appears to be reflection of $\displaystyle f(x)$ with respect to the line $\displaystyle y=x$ as mirror.I am not sure about this though.
I am looking for verification
You are right, I was not thinkin clearly. But I do not think that $\displaystyle f^{-1}(x)=x-\sin(x)$, because for example $\displaystyle f(f^{-1}(2))\ne{2}$