Sums of even and odd functions

In summary, the conversation discusses the rewriting of a function as the sum of an even and an odd function. It includes a proof that the given function k is neither even nor odd on any interval (-a,a), finding an even and an odd function to represent k, and an explanation of why k cannot be expressed as the sum of the two functions for all values in its domain.
  • #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
 
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  • #2
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
 

What are even and odd functions?

Even and odd functions are two types of functions that have specific properties. Even functions have the property that f(-x) = f(x), meaning that they are symmetric about the y-axis. Odd functions have the property that f(-x) = -f(x), meaning that they are symmetric about the origin.

What is the sum of an even and odd function?

The sum of an even and odd function is always an odd function. This is because when you add an even and odd function together, the even terms will cancel out and the remaining terms will be odd.

Can the sum of an even and odd function be an even function?

No, the sum of an even and odd function can never be an even function. This is because the even terms will always cancel out and leave an odd function as the result.

How can the sum of even and odd functions be useful?

The sum of even and odd functions can be useful in simplifying complex functions. By breaking down a function into its even and odd components, it can be easier to analyze and understand the behavior of the function.

Are there any real-world applications of even and odd functions?

Yes, even and odd functions are commonly used in physics and engineering to model symmetric and anti-symmetric systems. They are also used in signal processing and image recognition algorithms.

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