Sums of even and odd functions

  • #1

Homework Statement



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


Homework Equations



N/A

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,
Charismaztex
 

Answers and Replies

  • #2
Gib Z
Homework Helper
3,346
5
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
 

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