Question

In 1A-4 B, we are trying to show that an even polynomial = even + odd functions. However the starting function on the problem sheet is different than the starting one on the answer sheet. Is this a mistake or an extra step?

Answer 1

The equation for f(x) in 1A-4 (b) has a typo. They wanted to write an identity, which we can derive. Begin by writing f(x) this way:
\[ f(x)= \frac{f(x)}{2} + \frac{f(x)}{2} \]
and then add and subtract f(-x)/2 (which adds zero):
\[ f(x)= \frac{f(x)}{2} + \frac{f(x)}{2} + \frac{f(-x)}{2} - \frac{f(-x)}{2} \]
and reorder the terms to get
\[ f(x)= \frac{f(x)}{2} + \frac{f(-x)}{2}+ \frac{f(x)}{2} - \frac{f(-x)}{2} \\
f(x)= \frac{f(x)+f(-x)}{2} + \frac{f(x)-f(-x)}{2}
\]
if we simplify the right-hand side we get f(x), so the identity is clearly true.
But as the answer shows, the two terms represent even and odd functions that add up to create f(x)