# Even, Odd, or Neither

I would like someone to verify my solutions to the problems above?

9a. even 9b. odd 10. neither

• Correct. ${{{}}}$ Dec 12, 2013 at 18:55
• Even functions are symmetrical with regards to the vertical axis Oy, and odd functions are symmetrical with regards to the point of origin, O. Dec 13, 2013 at 4:01

Yes, you are right.

### Proper explanation

We say function $f$ is even, if for each $x \in D_1$, where $D_1$ is the domain of the function $f$, the following condition is satisfied: $$f(x) = f(-x).$$

On the other hand, we say, that function $g$ is odd, if for each $x \in D_2$, where $D_2$ is the domain of the function $g$, the following condition is satisfied: $$- g(x) = g(-x)$$ That's exactly the same as the fact, that $f$ is symmetric about the $y$-axis, resp. that $g$ is symmetric about the origin point in the coordinate system (simply the point $[0,0]$). But usually I assume you won't be given the graph of the function, or you would not be able to draw the graph of the function precisely.

One of the easiest examples are:

even function $f(x) = \vert\, x\,\vert$, where $|\cdot|$ is absolute value.

odd function $g(x) = x$.

Task. Of course not every function is even or odd, but there is one function which is both even and odd. Try to find it!

• You are right... I was oversimplifying that in my answer. Dec 12, 2013 at 19:49
• It is often better to show the simpleness. But since it was already answered I could afford to give a deeper explanation. I like to visualize mathematics a lot, but as you learn more it is harder to imagine everything and you won't avoid using strictly the definitions/theorems. Dec 12, 2013 at 19:56