Is the Function f(x) = (2x^2-x)/(x^2+x) Even, Odd, or Neither?

[Nitrate]

Homework Statement

is the function f(x) = (2x^2-x)/(x^2+x) even, odd, or neither?

Homework Equations

f(-x)=f(x) = even
f(-x)=-f(x) = odd
f(-x)≠f(x)≠ -f(x)

The Attempt at a Solution

f(x) = (2x^2-x)/(x^2+x)
f(-x)=(2(-x)^2+x)/((-x)^2+(-x))
f(-x) = (2x^2+x)/(x^2-x)

i think that's the right way to do it, but i don't know if it's even or odd.

 

[CompuChip]

That is the right way to do it.
So you have found the explicit form of f(-x).
Now is that equal to f(x), to -f(x), or neither?

 

[ehild]

Just try to substitute some value for x, say x=2 and x=-2. If f(2) is not equal either to f(-2) or -f(-2) than the function is neither odd nor even.

ehild

 

[Nitrate]

judging that the signage is switched from the original function to the f(-x) and the square terms stayed the same, then the function is even?

 

[Nitrate]

Just try to substitute some value for x, say x=2 and x=-2. If f(2) is not equal either to f(-2) or -f(-2) than the function is neither odd nor even.

ehild

never saw it that way. thanks :)

 

[HallsofIvy]

While you can use that "counter-example" method to prove that a function is neither even nor odd (and most functions are), you cannot use it to prove a function is either even or odd. The fact that f(2)= f(-2) does NOT prove it happens for all x.