Question

Could someone please explain problem 1A-4 parts a, b and c in problem set 1. Thanks.

Answer 1

a) Think of a polynomial as a sum of its components with even powers and those with odd powers.
b) There might be a mistake there. All other sources I check state that the second function should be 1/2(f(x)-f(-x)). If you have that then adding the fractions cancels out to give you f(x). To prove that the functions are even or odd take each of them separately. For example, let a(x)=1/2(f(x)+f(-x)). Find a(-x) and use the definition of odd and even to test.
c) Simply substitute 1/(x+a) for f(x) and 1/(-x+a) for f(-x), simplify and you will get a sum of an odd function and an even function that still equals 1/(x+a)

Answer 2

thank you so much!