How do you know if # f(x)=x^3+1# is an even or odd function?
Odd.
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Neither.
This function is neither odd nor even.
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To determine if the function ( f(x) = x^3 + 1 ) is even or odd, we need to evaluate whether ( f(x) = f(x) ) for all ( x ) (even function) or ( f(x) = f(x) ) for all ( x ) (odd function).
Let's check:
 For even functions: ( f(x) = f(x) ) [ f(x) = (x)^3 + 1 = x^3 + 1 ] [ f(x) = x^3 + 1 ]
Since ( f(x) ) is not equal to ( f(x) ), ( f(x) = x^3 + 1 ) is not an even function.
 For odd functions: ( f(x) = f(x) ) [ f(x) = (x)^3 + 1 = x^3 + 1 ] [ f(x) = (x^3 + 1) = x^3  1 ]
Since ( f(x) ) is not equal to ( f(x) ), ( f(x) = x^3 + 1 ) is not an odd function.
Therefore, ( f(x) = x^3 + 1 ) is neither an even nor an odd function.
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To determine if the function ( f(x) = x^3 + 1 ) is even or odd, we need to examine its symmetry properties under reflection about the yaxis and the origin.

Even functions satisfy the property ( f(x) = f(x) ) for all ( x ) in their domain. If ( f(x) = f(x) ), then the function is even.

Odd functions satisfy the property ( f(x) = f(x) ) for all ( x ) in their domain. If ( f(x) = f(x) ), then the function is odd.
For ( f(x) = x^3 + 1 ):

Test for evenness: ( f(x) = f(x) ): ( (x^3 + 1) = ((x)^3 + 1) ) This simplifies to: ( x^3 + 1 = x^3 + 1 ) As we can see, the function is not even because the lefthand side does not equal the righthand side.

Test for oddness: ( f(x) = f(x) ): ( (x^3 + 1) = ((x)^3 + 1) ) This simplifies to: ( x^3 + 1 = (x^3 + 1) ) This simplifies further to: ( x^3 + 1 = x^3  1 ) This is also not true, so the function is not odd either.
Therefore, ( f(x) = x^3 + 1 ) is neither even nor odd.
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When evaluating a onesided 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 onesided 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 onesided 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 onesided 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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