# How do you know if #f(x) =-x^3 + 6x# is an even or odd function?

Odd.

Here,

Thus, the function is odd. Notice how the function is comprised totally of odd-powered variables.

graph{-x^3+6x [-16.02, 16.02, -8.01, 8.01]}

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To determine if a function ( f(x) ) is even or odd, we examine its symmetry properties:

- Even functions satisfy ( f(-x) = f(x) ) for all ( x ) in their domain.
- Odd functions satisfy ( f(-x) = -f(x) ) for all ( x ) in their domain.

For ( f(x) = -x^3 + 6x ), we evaluate ( f(-x) ) and ( -f(x) ):

- ( f(-x) = -(-x)^3 + 6(-x) = -(-x^3) - 6x = -(-x^3) - 6x = -x^3 - 6x )
- ( -f(x) = -(-x^3 + 6x) = x^3 - 6x )

Comparing ( f(-x) ) and ( -f(x) ):

- Since ( f(-x) = -x^3 - 6x ) and ( -f(x) = x^3 - 6x ), the function does not satisfy the condition for even functions (( f(-x) \neq f(x) )).
- Since ( f(-x) = -x^3 - 6x ) and ( -f(x) = x^3 - 6x ), the function does not satisfy the condition for odd functions (( f(-x) \neq -f(x) )).

Therefore, ( f(x) = -x^3 + 6x ) is neither an even nor an odd function.

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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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