Determine if the function is odd, even, or neither in order to find the symmetry.

1. If odd, the function is symmetric about the origin.

2. If even, the function is symmetric about the y-axis.

Find by substituting for all occurrence of in .

Simplify the numerator.

Rewrite as .

Since both terms are perfect cubes, factor using the difference of cubes formula, where and .

Simplify.

Apply the product rule to .

Raise to the power of .

Multiply by .

Multiply by .

Raise to the power of .

Simplify the denominator.

Let . Substitute for all occurrences of .

Factor out of .

Factor out of .

Factor out of .

Factor out of .

Replace all occurrences of with .

Simplify terms.

Move the negative in front of the fraction.

Factor out of .

Rewrite as .

Factor out of .

Rewrite as .

Factor out of .

Rewrite as .

Factor out of .

Rewrite as .

Cancel the common factor.

Rewrite the expression.

Check if .

Since , the function is not even.

The function is not even

The function is not even

Find .

Simplify the numerator.

Rewrite as .

Since both terms are perfect cubes, factor using the difference of cubes formula, where and .

Simplify.

Move to the left of .

Raise to the power of .

Factor out of .

Factor out of .

Factor out of .

Factor out of .

Since , the function is not odd.

The function is not odd

The function is not odd

The function is neither odd nor even

Since the function is not odd, it is not symmetric about the origin.

No origin symmetry

Since the function is not even, it is not symmetric about the y-axis.

No y-axis symmetry

Since the function is neither odd nor even, there is no origin / y-axis symmetry.

Function is not symmetric

Find the Symmetry f(x)=(x^3-27)/(x^2-6x)