Question

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I'm having trouble wrapping my head around 1A-4, but I think I found the problem. First of all, the actual exercise version has a typo: the last term should have a minus between the numerator terms.

The solution writes this correct, but gets G(-x) wrong as far as I can see. It should read:
\[G(-x)=\frac{f(-x)-f(x)}{2}=-\frac{f(x)-f(-x)}{2}=-G(x)\]

Answer 1

Yes, you are correct, and they muddled both the question and the answer.
https://en.wikipedia.org/wiki/Even_and_odd_functions
has some simple examples.

If they start with the correct expression:
\[ f(x) = \frac{ f(x) + f(-x)}{2} + \frac{ f(x) - f(-x)}{2} \]
It is easy to add the fractions on the right-hand side to get
\[ f(x) = \frac{ f(x) + f(-x)+f(x) - f(-x)}{2} = \frac{2f(x)}{2}= f(x)\]
showing the identity is correct.

If we define the first fraction as
\[ F(x) = \frac{ f(x) + f(-x)}{2} \]
we see
\[ F(-x) = \frac{ f(-x) + f(-(-x))}{2} = \frac{ f(-x) + f(x)}{2} = F(x)\]
which shows F(x) is even

similarly, define the second fraction as
\[ G(x) = \frac{ f(x) - f(-x)}{2} \]
and we find
\[ G(-x) = \frac{ f(-x) - f(x)}{2} = - \frac{ f(x) - f(-x)}{2} = -G(x)\]
which shows G(x) is odd.
(See wikipedia link for details on even/odd )

Answer 2

I struggled with this for a while until I realized that they had messed up the question.