How do you determine whether a function is odd, even, or neither: #h(x)= -x^3/(3x^2-9)#?

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Answer 1 · 2017-07-20T13:26:18

Odd

We substitude #x# with #-x# :

#h(-x)=-(-x)^3/(3(-x)^2-9)=-(-x^3)/(3x^2-9)=x^3/(3x^2-9)=#

#-h(x)#

Now because #h(-x)=-h(x)# the function is odd.

Answer 2 · 2017-07-20T13:35:43

#h(x)# is an odd function.

We use the following condition:

# { (f(-x)=f(x)), (f(-x)=-f(x)) :} => {: (f " is even"), (f " is odd") :}#

So for the given function:

# h(x) = -(x^3)/(3x^2-9) #

And so:

# h(-x) = -((-x)^3)/(3(-x)^2-9) #

# " " = -(-x^3)/(3x^2-9) #

# " " = (x^3)/(3x^2-9) #

# " " = -h(x) #

And we conclude that #h(x)# is an odd function.

We can verify this graphically, as odd functions have rotational symmetry about the origin:
graph{-(x^3)/(3x^2-9) [-20, 20, -10, 10]}