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Show that $y+sin(y) = x$ in a neighborhood of $(0,0)$ can be written as function of $x$. I'm not sure I understand the question. In class we learned the inverse function theorem and the implicit function theorem but I'm a little confused. Any help would be appreciated!

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Let $$F(x,y)=y+\sin[y]-x$$ and try to calculate $$F_{y},F_{x}$$ at point $(0,0)$. Then apply implicit function theorem. You may think this as an ODE, where you have $$ F(x,y)=0\rightarrow y'=\frac{dy}{dx}\sim -\frac{F_{x}}{F_{y}} $$ and you want to solve it by locally construct $y(t)-y(t_{0})=\int^{t}_{x_{0}}y'(s)ds$. This is not rigorous but might be helpful.

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You want to show that the equation $y + \sin y = x$ defines a function of $x$ in a neighborhood of $(0,0)$. The implicit function theorem is what you need. Show that the hypotheses of the implicit function theorem are satisfied for your case.

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