find $f\circ f$ for the function $f\colon \mathbb R^2\to \mathbb R^2$ (,)=(−,) I know that if (,)=(−,), then () is its inverse reflected about the -axis. If this is the case then $f\circ f$ = f^-1(−f^-1(−)). I also know that it may equal (−,−) but I have no idea how (−,)=(−,-). I also know that its got something to do with vectors or scalars but I'm still stuck. I need someone to explain it in detail for me.
I am not sure on how to do this question. Could someone please help me?
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I think you are confused because you have $x$ and $y$ in the definition of $f$ and the argument of $f \circ f$, so it might help to replace it somewhere.
So $f$ is a function that maps $(x, y)$ to $(-y,x)$. To get $f \circ f$, we need to apply $f$ again. Now let $u=-y$ and $v=x$. From the definition, we have that $$f(u,v)=(-v,u)=(-x,-y)$$ So $$(f \circ f)(x,y)=(-x,-y)$$
If I get your question right, you just want to know what $f \circ f$ is for your function. Observe that $(f \circ f)(x,y) = f(f(x,y)) = f(-y, x) = (-x, -y)$.