Taking Image of a curve about a given line

In summary, finding the image of a function about a given line involves reflecting the function around the line. This can also be done with functions, but they may not always be explicitly solvable. The functions $f(x)= x + sinx$ and $f(x)= y + sin y$ are symmetric around $y=x$ and intersect at the point $(2\pi,2\pi)$. The area under $g$ between the limits is $2\pi^2$, resulting in an answer of $2$.
  • #1
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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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  • #2
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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