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

9a. even 9b. odd 10. neither

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

1 Answer 1

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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!

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  • $\begingroup$ You are right... I was oversimplifying that in my answer. $\endgroup$
    – apnorton
    Dec 12, 2013 at 19:49
  • $\begingroup$ 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. $\endgroup$
    – quapka
    Dec 12, 2013 at 19:56

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