Understanding Even and Odd Functions: Exponents and Symmetry Tests Explained

[math nerd]

Can you always tell if a function is odd or even by looking at the exponents of each of the variables? My book says you can but when I look in other books it gives examples when that is not true. Or do you always have to do symmetry tests to decide?

 

[HallsofIvy]

It doesn't help to post twice! I will delete the other thread.

The answer to your question is "yes and no"!

IF the function in question is a polynomial then yes: a function is "even" if and only if all powers of x are even. A function is "odd" if and only if all powers of x are odd. That should be easy to remember! I suspect your book is only talking about polynomials.

However, the concepts of "even" and "odd" functions apply to all functions, not just polynomials. A function is called "even" if f(-x)= f(x) for all numbers x (changing the sign on x doesn't change the value of the function at all), odd if f(-x)= -f(x) (changing the sign on x only changes the sign on f(x)) for all numbers x. Of course, with most functions there is no simple relation between f(x) and f(-x): most functions are neither even nor odd.

f(x)=cos(x) for example is an even function while g(x)= sin(x) is an odd function.
f(x)= x+ 3 is neither even nor odd.

 

[Architeuthis Dux]

I understand the confusion around determining whether a function is odd or even based on its exponents. While it may seem straightforward at first, there are certain cases where this method may not apply. This is because the parity (odd or even) of a function is determined by its behavior when the input is changed to its opposite, or negative value.

In some cases, the exponents of the variables may give us a hint about the function's symmetry, but it is not always a reliable method. This is why it is important to also perform symmetry tests to confirm the parity of a function. These tests involve substituting the input with its opposite and seeing if the function remains unchanged or if it changes sign.

Additionally, it is important to note that there are other types of symmetry tests, such as graphical symmetry, that can also be used to determine the parity of a function. These tests may be more reliable in certain cases where the exponents method may not work.

In summary, while the exponents of a function may provide some insight into its parity, it is not always a definitive method. It is important to also perform symmetry tests to accurately determine the parity of a function. As a scientist, it is crucial to use multiple methods and approaches to validate our findings and ensure accuracy in our understanding.