# How do you determine if #f(x) = x^3 - x^7# is an even or odd function?

odd function

Examine the following to see if a function is even or odd.

• f(x) is even if f(x) = f( -x)

About the y-axis, even functions are symmetrical.

• f(x) is odd if f(-x) = - f(x).

There is symmetry around the origin of odd functions.

Check for even

f(x) is not even since f(x) ≠ f(-x).

Check for odd

f(x) is odd since f(-x) = - f(x).

See the symmetry surrounding the origin in this graph of f(x): graph{x^3-x^7 [-10, 10, -5, 5]}

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To determine if ( f(x) = x^3 - x^7 ) is an even or odd function:

- Even Function: If ( f(-x) = f(x) ) for all ( x ) in the domain, then the function is even.
- Odd Function: If ( f(-x) = -f(x) ) for all ( x ) in the domain, then the function is odd.

Apply these conditions to ( f(x) = x^3 - x^7 ):

- Substitute ( -x ) into ( f(x) ) to test for evenness.
- Substitute ( -x ) into ( -f(x) ) to test for oddness.
- Check if either condition holds true for all ( x ) in the function's domain.

After these tests, determine whether the function is even, odd, or neither.

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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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