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How do you determine whether a function is odd, even, or neither: #f(x)=2x^4-x^2#?

Answer 1

Benjamin Achtenberg

even

even function #f(x) = f(-x)#

odd function #f(-x) = - f(x) #

here #f(-x) = 2 (-x)^4 - (-x)^2 = 2x^4 - x^2 = f(x)#

ie even

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Answer 2

Theodore Amidon

To determine whether a function is odd, even, or neither, we check how the function behaves under the following transformations:

Odd Function: ( f(-x) = -f(x) ) for all ( x ) in the domain.
Even Function: ( f(-x) = f(x) ) for all ( x ) in the domain.
Neither Odd nor Even: If the function does not satisfy either of the above conditions.

For the function ( f(x) = 2x^4 - x^2 ):

Odd Function: ( f(-x) = 2(-x)^4 - (-x)^2 = 2x^4 - x^2 = f(x) )
Since ( f(-x) = f(x) ), the function is even.

Therefore, the function ( f(x) = 2x^4 - x^2 ) is an even function.

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Answer from HIX Tutor

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.