# Thread: Determining whether functions are odd, even or neither

1. ## Determining whether functions are odd, even or neither

How do you do this without solving algebraically?

Just by looking at the equation how do you tell which one it is?

for example is the following function even, odd or neither?

y=x^3/(x^5-x^2)

explanations appreciated

2. Originally Posted by smiley_x
How do you do this without solving algebraically?

Just by looking at the equation how do you tell which one it is?

for example is the following function even, odd or neither?

y=x^3/(x^5-x^2)

explanations appreciated
$\displaystyle f(x) = \frac{x^3}{x^5 - x^2} = \frac{x}{x^3 - 1}$, $\displaystyle x \neq 0$.

Now note that $\displaystyle f(-x) = \frac{(-x)}{(-x)^3 - 1} = \frac{x}{x^3 + 1}$. Since this is not equal to f(x) and not equal to -f(x) the function is neither even nor odd.

3. Hello, smiley_x!

How do you do this without solving algebraically?

Just by looking at the equation how do you tell which one it is?

For example, is the following function even, odd or neither?

. . $\displaystyle y\:=\:\frac{x^3}{x^5-x^2}$
If the function is composed of polynomials only, there is an "eyeball" rule.

. . If all the exponents of $\displaystyle x$ are even, it is an even function.
. . If all the exponent of $\displaystyle x$ are odd, it is an odd function.

Note that a constant is an even term,
. . because $\displaystyle 5 \:=\:5x^0$ has an even exponent.

The given function has both even and odd exponents.
. . It is a Neither.