Problem set 1, 1a-4 (b), is there a typo in the formula provided? The problem set has the numerator of second function on the right as f(x)+f(-x) whereas the solutions have it as f(x)-f(-x). I get that if the second function is an odd function then f(-x) = -f(x), but I'm getting myself tied in knots in terms of how that fits into the overall function provided. Thx!

Question

Problem set 1, 1a-4 (b), is there a typo in the formula provided?  The problem set has the numerator of second function on the right as f(x)+f(-x) whereas the solutions have it as f(x)-f(-x).  I get that if the second function is an odd function then f(-x) = -f(x), but I'm getting myself tied in knots in terms of how that fits into the overall function provided.  Thx!

anonymous · Answer

There is a typo in the question.  It should read:

$$f(x) = \frac{ f(x) + f(-x) }{ 2 } + \frac{ f(x) - f(-x)}{ 2 }$$

If you add up the right side you will see that that this comes out to $$\frac{ f(x) + f(x) }{ 2 } = f(x)$$
which is equal to the left side.

The solution states the problem correctly.  However, the solution also contains a typo.  The last line of the solution should read:

$$G(-x) = \frac{ f(-x) - f(-(-x)) }{ 2 } = \frac{ f(-x) - f(x) }{ 2 } = - G(x)$$

I have encountered quite a few typos and discrepancies between the problems and solutions for this problem set.

anonymous · Answer

perfect - thanks!