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I have the problem.

Let $$f(t) = \frac{t - x}{t + y}.$$

Show that $$f(x + y) + f(x - y) = \frac{-2y^2}{x^2 + 2xy}.$$

I know that this is just some substitution followed by simplification, but am missing the point of the let $f(t) = (t - x)/(t + y)$. It doesn't seem to fit into the "show that" portion of the question. Can someone steer me in the right direction?

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It just defines what $f$ means in the next line. $x$ and $y$ must be presumed to be constants that don't change during the entire problem (since they are free in the definition of $f$ and have not previously been assigned a meaning). – Henning Makholm Oct 11 '11 at 21:36
up vote 4 down vote accepted

The quantities $x$ and $y$ are some fixed quantities. Given any number $a$, $$f(a) = \frac{a-x}{a+y}.$$ Now, what happens if $a$ is equal to $x+y$? Will $$f(x+y) = \frac{(x+y) - x}{(x+y) + y} = \frac{y}{x+2y}.$$

Similarly, what happens if $a$ is $x-y$? You plug in $x-y$ for $t$ into the formula.

So, plug in $x+y$ for $t$; plug in $x-y$ for $t$. Add the results. See if you get what you are supposed to get.

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Thank you, the problem appears so much clearer when elaborated in this fashion. – erimar77 Oct 11 '11 at 21:38
Arturo has that effect on lots of people! – The Chaz 2.0 Oct 11 '11 at 21:39
Also, thanks to everyone for properly formatting my request, it was most appreciated! – erimar77 Oct 11 '11 at 21:46

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