1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Sums of even and odd functions

  1. Mar 1, 2010 #1
    1. The problem statement, all variables and given/known data

    If f:(-a,a)-->Real numbers, then f can be rewritten as the sums of an even and an odd function

    Let k: Real numbers\{-1}-->Real numbers be given by [tex]k(x)=\frac{x^2+4}{x+1}[/tex]

    (i) Prove that there is no interval (-a,a) on which k is either even or odd
    (ii) Find an even function g, an odd function h and a value a for which
    [tex]k(x)= g(x) +h(x)[/tex], x belongs to (-a,a)
    (iii) Explain why it is, or is not, true that [tex]k(x)=g(x) +h(x)[/tex] for all x in the domain of k

    2. Relevant equations


    3. The attempt at a solution

    (i) Is it true that by proving k(x) does not equal k(-x) and k(x) does not equal -f(-x), we can prove that there is no interval (-a,a) on which k is either even or odd?

    (ii) I am not quite sure here.

    (iii) is it not true because x cannot be -1?

    Thanks in advance,
  2. jcsd
  3. Mar 3, 2010 #2

    Gib Z

    User Avatar
    Homework Helper

    For i) use a proof by contradiction. Assume there is an interval (-a,a) on which k is either even or odd, then on some subset of that, say [-b,b], it must be either even or odd as well.

    ii) Study t(x)= f(x) - f(-x)

    iii) Give it another look once you find g(x) and h(x). In response to your original attempt, I suggest you read the question again carefully. Whilst it is true that x can not be -1, -1 is not in the domain of k. Good luck
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook