Symmetric functions/odd even or neither

[schlynn]

Homework Statement

I am supposed to find out if this function is symmetric with reference to the y-axis or reference to the origin.
The function is
F(x)=4x^2/(x^3+x)

Homework Equations

These are how to know if even or odd
A(-x)^even power = ax^even power
A(-x)^odd power = -ax^odd power

The Attempt at a Solution

I think that to solve these u change x to -x in the function right? So I got...
F(-x)=4-x^2/(-x^3-x)
To
F(-x)=4+x/(-x-x)
To
F(-x)=4+x/-2x
To
F(-x)=2+x/-x
To
-2

But now what, how do I tell if it's odd even or neither?? Did I even do the math right?

 

[Mark44]

Homework Statement

I am supposed to find out if this function is symmetric with reference to the y-axis or reference to the origin.
The function is
F(x)=4x^2/(x^3+x)

Homework Equations

These are how to know if even or odd
A(-x)^even power = ax^even power
A(-x)^odd power = -ax^odd power

These equations aren't what you use to determine the evenness or oddness of a function.
If f(-x) = f(x) for all x in the domain of f, f is an even function.
If f(-x) = -f(x) for all x in the domain of f, f is an odd function.

The Attempt at a Solution

I think that to solve these u change x to -x in the function right? So I got...
F(-x)=4-x^2/(-x^3-x)

No, you need some parentheses here.
F(-x) = 4(-x)^2/((-x)^3 + (-x))
= 4x^2/(-x^3 - x)
= -(4x^2)/(x^3 + x)
= ?

To
F(-x)=4+x/(-x-x)
To
F(-x)=4+x/-2x
To
F(-x)=2+x/-x
To
-2

What are you doing here? You can't get the first of the four lines from what you started with, and the third line doesn't follow from the second, and the fourth doesn't follow from the third.

But now what, how do I tell if it's odd even or neither?? Did I even do the math right?

 

[ravx]

In general, a function is Even if
$$F(x) = F(-x)$$
and a function is Odd if
$$F(x) = -F(-x)$$

So consider in your case
$$F(x) = -\frac{x^2}{x^3+x}$$

Look now at $$F(-x)$$:

$$F(-x) = -\frac{(-x)^2}{(-x)^3+(-x)}$$
$$= -\frac{x^2}{-x^3-x}$$
$$= \frac{x^2}{x^3+x}$$
$$= -F(x)$$

So your function is Odd.

Another approach is to treat it as a product or quotient of even or odd functions

Odd function * Odd function = Even function
Even * Even = Even
Even * Odd = Odd
Odd * Even = Odd

Likewise
Odd / Odd = Even
Even / Even = Even
Even / Odd = Odd (this is your case)
Odd / Even = Odd