Question

Odd/even questions: a) Show that every polynomial is the sum of an even and an odd function.

I'm confused about this because the lectures only briefly mentioned odd vs even functions. I still want to solve the problem, so how would I approach this?

Answer 1

I'm also confused about part (b).

Generalize part (a) to an arbitrary function f(x) by writing
f(x) = (f(x) + f(−x))/2 + (f(x) + f(−x))/2
Verify this equation, and then show that the two functions on the right are respec­tively even and odd.

Is there a typo here? Aren't the two functions on the right exactly the same?

Answer 2

And isn't part (b) the same as part (a)? (b) can't be a generalization of part (a) because part (a) is not specific at all. It looks like (b) is just the procedure for proving that any polynomial is the sum of an odd and even function.

Answer 3

split the given polynomial into two functions. One consisting of even powers of x, and one consisting the odd powers of x.

Answer 4

That's my problem. They don't give me a polynomial. They say to prove it for all polynomials. And they tell me how to in part (b), but I think there's a typo because both of the terms look the same. (One is supposedly odd, and the other even). See my second post.

Answer 5

yes there is a typo. in part (b), f(x)=(f(x)+f(-x))/2 + (f(x)-f(-x))/2 is the right question.

Answer 6

Oh! That's what I was guessing, but I couldn't be sure. Thanks! And (a) and (b) are the same question, right?

Answer 7

Any polynomial has this general form. \[f(x)=\sum_{0}^{\infty}a _{k}x ^{k}\]

Answer 8

So write this as two summations of even and odd powers of x. The two functions obtained will therefore be even and odd respectively.

Answer 9

Thank you for your quick replies and your clear explanation.