How do you know if # f(x)=x^3+1# is an even or odd function?

Answer 1

Odd.

Quite simply, whether a function is even or odd is whether the degree (the largest exponent) is even or odd. The degree of #x^3 + 1# is #3#, which is odd, making this an odd function.
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Answer 2

Neither.

An function is even if: #f(-x)=f(x)#
A function is odd if: #f(-x)=-f(x)#
If #f(-x)=x^3+1#, the function is even. If #f(-x)=-x^3-1#, the function is odd.
So, find #f(-x)#.
#f(-x)=(-x)^3+1#
#f(-x)=-x^3+1#

This function is neither odd nor even.

A good way to check is by recognizing that even functions are reflections of themselves over the #y#-axis and odd functions are reflections of themselves over the #x#-axis.
This is the graph of #x^3+1#: graph{x^3+1 [-10, 10, -5, 5]}
Notice that if the graph were shifted down one unit (the function #x^3#), the graph would be a reflection of itself over the #x#-axis and would indeed be odd.
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Answer 3

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:

  1. 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.

  1. 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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Answer 4

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 y-axis and the origin.

  1. 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.

  2. 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 left-hand side does not equal the right-hand 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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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.

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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