Figuring Out an Odd Function With Different Parts Along x-Axis: Help Needed!

[TSN79]

I just can't seem to grasp this! I have no problems finding out if a function let's say $$x-2x^2$$ is an even or odd function, but when the function is defined differently along different part along the x-axis then I don't understand anything! This function:
$$f(x)=\left\{\begin{array}{cc}0 &\mbox{ if }<br /> -2\leq x<0\\(1/2)x & \mbox{ if }0\leq x<2\end{array}\right$$

Someone help me please!
This function is supposed to be neither acutally, but I have no idea how to show this...

 

[arildno]

Well let x=1
Then f(1)=1/2, but f(-1)=0 which is not equal to either 1/2 or -1/2.
Hence, f(x) is neither even nor odd.

 

[Gokul43201]

Extra Note : Keep in mind that the above method of comparing f(a) with f(-a) for a particular choice of 'a', can be used only to show that f is neither even nor odd.

To show that some f is even or odd in a given domain, you must show that the relevant relationship holds for all 'a' in the specified domain.

 

[arildno]

As Gokul said, I gave a SUFFICIENT proof of f being neither even or odd, by providing a COUNTER-EXAMPLE (of even-ness and odd-ness).

 

[TSN79]

As Gokul said, it only gives me the answer at the point a. I can show that each of the functions separately are either even or odd (or neither), but how do I show this for a given domain...? I know the definition for an odd function is f(-x) = -f(x), and for an even function f(-x) = f(x), but in what function should I put in the negative x?? I have two (sometimes more) to chose from, 0 and (1/2)x. Help! Nå ser jeg jo at jeg sikkert kunne skrevet norsk her også...

 

[arildno]

It's enough with a single counter-example to prove that it is neither even or odd on the given domain (the condition for even-ness must hold for ALL members in the domain in order for the function to be even).