What is an odd function?

A function is said to be ODD, when for every `x` value belonging to its domain, the symmetric of `x` has an image which is also symmetric. According to mathematical language we’ll have: `-f(x) = f(-x) , x in D_f`. For those who do not understand about the subject, it may seem a little confusing. Therefore, I’ll give an example to help you to understand.

This graph shows a cubic function (`y=x^3`) which is an odd function. Here we can notice that object 2 has an image 8 and its symmetric, which is object -2 having an image -8. According to mathematical language it means: `f(2) = 8 ^^ f(-2) = -8`. We have to pay attention to the definition, according to which it is not enough the existence of two symmetric objects having the corresponding images as being symmetric! For a function to be considered odd, all the objects and corresponding symmetric must have symmetric images.

Thus, these functions are peculiar since they are symmetric regarding the origin of the referential. In order to be able to “see” it, we have to fold a sheet twice in which the graph is shown. While the first fold is vertical. The second one will be horizontal. If the function is odd, both parts of the graph, which can be found in opposite quadrants, will lay over each other.

If a function is not odd, is it even?

That is a common mistake made by pupils: If a number is not odd, it is obviously even. So, if a function is not odd, it is even for sure.” However, concerning functions, this is not true. There is a completely different definition for even functions, that is explained in another topic.

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