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I am trying to solve this using what I have read from this site but I always hit a dead end. Consider the function $$f: \mathbb R \times \mathbb R \to \mathbb R\times\mathbb R$$ defined by $$f(x,y) = (x+y, x-y)$$ Show that the inverse is $$f^{-1}(a,b) = \left(\frac{a+b}2, \frac{a-b}2\right),\forall (a,b)\in\mathbb R \times\mathbb R$$.

Adi Dani
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Alex Martin
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  • So what happens when you are given $x, y$ and you plug them in to the function $f\circ f^{-1}(x,y)$? – abiessu Nov 14 '13 at 00:05

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If $x+y=a,x-y=b$ then $x=\frac{a+b}{2},y=\frac{a-b}{2}$ and $$f\left(\frac{a+b}{2},\frac{a-b}{2}\right)=(a,b)$$ $$f^{-1}\left(f\left(\frac{a+b}{2},\frac{a-b}{2}\right)\right)=f^{-1}(a,b)$$ $$f^{-1}(a,b)=\left(\frac{a+b}{2},\frac{a-b}{2}\right)$$ because $$f\left(f^{-1}(a,b)\right)=(a,b)$$

Adi Dani
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