# How do you determine if #f(x)= (-x³)/(3x²-4)# is an even or odd function?

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OVERALL, TALKING _

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Function EVEN: f(-x) = f(x)

Function of ODD: f(-x) = - f(x)

THE QUESTION IN ACT

which is

This is peculiar.

_ A PICTURE _

graph{ (-x^3)/(3x^2-4) [-18.02, 18.02, -9.01, 9.01]}

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To determine if a function ( f(x) ) is even or odd, we need to check if it satisfies the following properties:

- Even Function: ( f(x) = f(-x) ) for all ( x ) in the function's domain.
- Odd Function: ( f(x) = -f(-x) ) for all ( x ) in the function's domain.

For the function ( f(x) = \frac{-x^3}{3x^2 - 4} ), we'll evaluate both ( f(x) ) and ( f(-x) ).

If ( f(x) = f(-x) ), then the function is even. If ( f(x) = -f(-x) ), then the function is odd.

Let's evaluate ( f(x) ) and ( f(-x) ):

[ f(x) = \frac{-x^3}{3x^2 - 4} ]

[ f(-x) = \frac{-(-x)^3}{3(-x)^2 - 4} = \frac{-(-x)^3}{3x^2 - 4} = \frac{-(-x^3)}{3x^2 - 4} = \frac{x^3}{3x^2 - 4} ]

Now, we'll compare ( f(x) ) and ( f(-x) ):

[ f(x) \neq f(-x) ]

[ f(x) \neq -f(-x) ]

Since neither ( f(x) = f(-x) ) nor ( f(x) = -f(-x) ) holds true for all ( x ) in the function's domain, the function ( f(x) = \frac{-x^3}{3x^2 - 4} ) is neither even nor odd.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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