What is an even function?

A function is even when every `x` value belongs to its domain and the `x` symmetric has the same image. According to mathematical language it means: `f(x) = f(-x) , x in D_f`. For those who don't understand the subject, it seems a little confusing. So, I'll explain it better for you to understand.

This graph shows a quadratic function (`y=x^2`) which is an even function. Here we can notice that both the object 2 and its symmetric have image 4. According to mathematical language: `f(2) = f(-2) = 4` . But we have to pay attention, since according to the definition it is not enough to have two symmetric objects with the same image! For a function to be considered even all the objects and the correspondent symmetric must have the same image.

These functions are particularly symmetric regarding the y-axis (`O_y`). If we fold the sheet vertically, we will notice that the graph, on the left side of the sheet, will lay over the graph on the right side.

If a function is not even, is it odd?

This is a frequent mistake made by pupils: “if a number is not even, it is obviously odd. So if a function is not even, it is odd for sure.” However, in the case of functions, this is not true. There is a completely different definition concerning an odd function, that will be explained in another topic.

Check out our List of Questions to get to know a little more about the most diverse topics related to mathematics. If you have any pertinent (math) question whose answer can not easily be found, send us an email on the Contact page with the question. We will be happy to respond. In the event that you detect any errors in our answers, do not hesitate to contact us!