Taking Image of a curve about a given line

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SUMMARY

The discussion focuses on finding the image of the function \( f(x) = x + \sin x \) about the line \( y = -x \). The reflection of this function around the line is expressed as \( x = y + \sin y \), which cannot be explicitly solved for \( y \). Additionally, it is established that the functions are symmetric around \( y = x \) and intersect at the point \( (2\pi, 2\pi) \). The area under the curve \( g \) between specified limits is calculated to be \( 2\pi^2 \), leading to a final answer of \( 2 \).

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  • Ability to solve equations involving implicit functions
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How to find image of $f(x)= x + sinx$ about the given line $y = - x$ .

Similarly can we take image of a function about a function? OR is it necessary about which we take image should be a point, line only?
 
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If $y=x+\sin x$, then the reflection around $y=-x$ is $-x=-y+\sin(-y)$ or $x=y+\sin y$. Although a function, this cannot be explicitly solved for $y$. However the two functions are also symmetric around $y=x$ and intersect at the point $(2\pi,2\pi)$. So the area under $g$ between your limits is $2\pi^2$ so your answer is $2$.
 

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