Odd and Even Functions: The Mystery of Their Division

Thread starter hyper
Start date Jul 5, 2008
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even Functions

In summary, odd and even functions are two types of functions in mathematics that describe the symmetry of a graph. An odd function has a graph that is symmetrical about the origin, while an even function has a graph that is symmetrical about the y-axis. To determine if a function is odd or even, a test can be used. These functions have properties and applications that make them useful in mathematical models and in solving equations. A function cannot be both odd and even, but it can be an "odd-even" function with a symmetrical graph about both the origin and the y-axis. Odd and even functions have real-life applications in various fields such as signal processing and electrical engineering.

Jul 5, 2008

hyper

Why does two odd functions divided by each other become an even function?

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Jul 5, 2008

Defennder

Homework Helper

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?

Jul 5, 2008

HallsofIvy

Science Advisor

Homework Helper

[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]

Jul 5, 2008

hyper

thanks!

Related to Odd and Even Functions: The Mystery of Their Division

1. What are odd and even functions?

Odd and even functions are two types of functions in mathematics that describe the symmetry of a graph. An odd function has a graph that is symmetrical about the origin, meaning that if you fold the graph in half along the y-axis, the two halves will match up. An even function has a graph that is symmetrical about the y-axis, meaning that if you fold the graph in half along the y-axis, the two halves will also match up.

2. How do you determine if a function is odd or even?

To determine if a function is odd or even, you can use the following test:
- For an odd function, if f(-x) = -f(x) for all x in the domain, then the function is odd.
- For an even function, if f(-x) = f(x) for all x in the domain, then the function is even.

3. What is the significance of odd and even functions?

The significance of odd and even functions lies in their properties and applications. Odd functions have a zero value at the origin, while even functions have a minimum or maximum value at the origin. This makes them useful in mathematical models and in solving equations. Additionally, odd and even functions are important in Fourier series and other areas of mathematics.

4. Can a function be both odd and even?

No, a function cannot be both odd and even. If a function satisfies the conditions for both an odd and an even function, it is considered to be neither odd nor even. This type of function is called an "odd-even" function and has a graph that is symmetrical about both the origin and the y-axis.

5. How can odd and even functions be used in real life?

Odd and even functions can be used to describe and model many real-life situations. For example, odd functions can be used to represent odd phenomena such as the behavior of a pendulum or the flow of alternating current. Even functions can be used to model even phenomena such as the shape of a bowl or the distribution of heat in a circular object. Additionally, odd and even functions are used in signal processing, electrical engineering, and other fields.