Odd and Even Functions: The Mystery of Their Division

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Discussion Overview

The discussion centers around the properties of odd and even functions, specifically exploring why the division of two odd functions results in an even function. The scope includes mathematical reasoning and conceptual clarification.

Discussion Character

  • Mathematical reasoning

Main Points Raised

  • One participant questions why the division of two odd functions results in an even function.
  • Another participant provides a reasoning approach by stating the definitions of odd functions and suggests dividing the functions to observe the outcome.
  • A mathematical expression is presented that demonstrates the relationship between the functions when evaluated at -x, leading to the conclusion that the quotient behaves as an even function.

Areas of Agreement / Disagreement

The discussion does not indicate any disagreement, but it remains focused on the reasoning behind the property without exploring alternative viewpoints or potential exceptions.

Contextual Notes

Assumptions about the functions being defined and the conditions under which they are considered odd are not explicitly stated. The discussion does not address any limitations or exceptions to the property discussed.

Who May Find This Useful

Readers interested in the properties of odd and even functions, particularly in the context of mathematical functions and their behaviors under division.

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Why does two odd functions divided by each other become an even function?
 
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Well suppose you have 2 odd functions f(x) and g(x). Since they are odd, f(x)= -f(-x) and similarly for g(x). Now divide f(x) by g(x) on both sides of the equation. What do you observe?
 
[tex]\frac{f}{g}(-x)= \frac{f(-x)}{g(-x)}= \frac{-f(x)}{-g(x)}= \frac{f(x)}{g(x)}= \frac{f}{g}(x)[/itex][/tex]
 
thanks!
 

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