How do I figure out step by step if this function is even, odd, or neither?
f(x)=(x)/(x^2+1)

Question

How do I figure out step by step if this function is even, odd, or neither?
f(x)=(x)/(x^2+1)

amistre64 · Answer

plug in x= -x; and if you get back f(x), its even
if you get back -f(x) its odd
if neither, then its neither

amistre64 · Answer

$$f(-x)=\frac{(-x)}{(-x)^2+1}=-\frac{x}{x^2+1}=-f(x)$$

anonymous · Answer

so.. f(-x)=(-x)/(-x^2+1)=(-x)/(-x^2+1) ?

amistre64 · Answer

so far yes, but dont forget to wrap it ias (-x)^2 + 1

anonymous · Answer

so after that what do I do?

amistre64 · Answer

when i first learned these, the replacing x by -x seemed wierd; so if you want to replace it by a different negative letter thats fine

amistre64 · Answer

look about 4 posts up ... i worked it out

anonymous · Answer

ohh and if it comes out positive its even if it comes out negative its an odd?

amistre64 · Answer

if it comes out exactly the same; its even; for example:  I know x^2 is even

f(-x) = (-x)^2 = x^2 ; since x^2 = x^2;  f(-x) = f(x)

amistre64 · Answer

i know f(x) = x is an odd function;

f(-x) = -x = -f(x)

anonymous · Answer

and in my problem; its an odd function because x resulted in (-x) even though x^2+1 came back the same.. (-x)^2+1=x^2+1.. so if my problem was x^2/x^2+1... that would be an even function?

amistre64 · Answer

correct

amistre64 · Answer

you want to look at the overall effect it has on the function; not just parts of it

amistre64 · Answer

if the overall function:  f(-x)  simplifies out to  f(x)  its even
if it simplifies out to the opposite of f(x):  also known as  -f(x), then its off

if you cant get it into either of these, then its neither of them ...

anonymous · Answer

okay.. so for f(x)=(x^2)/(x^4+1)...f(-x)=(-x)^2/(-x)^4+1=x^2/x^4+1=even function?

amistre64 · Answer

correct

anonymous · Answer

yay! thanks.

amistre64 · Answer

for rational expressions, after a few of them you can develop a sense of their properties; they mimic the +- products.

even/even = even
odd/even = odd
even/odd = odd
odd/odd = even

anonymous · Answer

Okay. I'll remember that :)

anonymous · Answer

@amistre64 ... would f(x)=x|x| be and odd function.. f(-x)=(-x)|-x|
=f(-x)=-x|x|

amistre64 · Answer

|x| is even
 x is odd

even * odd = odd

amistre64 · Answer

and yes, the more proofy way:

f(-x) = (-x)|-x| = - [x|x|] = -f(x)

anonymous · Answer

okay thank you!

amistre64 · Answer

youre welcome