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  • one year ago

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?

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  1. phi
    • one year ago
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    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)

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