# How do you determine if #f(x)=4x^3# is an even or odd function?

In our illustration:

There is a quick way to determine if a polynomial is odd or even:

Do any terms have an odd degree, an even degree, or a combination?

If the function is odd, then it is odd; if it is even, then it is even; and if it is neither, then it is neither.

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odd function

Examine the following to see if a function f(x) 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 function.

f(x) is not even since f(x) ≠ f(-x)

Check for strange functions

f(x) is odd since - f(x) = f( -x) graph{4x^3 [-10, 10, -5, 5]}

Take note of the origin's symmetry.

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To determine if a function ( f(x) = 4x^3 ) is even or odd:

- Even function: ( f(x) ) is even if ( f(-x) = f(x) ) for all ( x ).
- Odd function: ( f(x) ) is odd if ( f(-x) = -f(x) ) for all ( x ).

Let's evaluate ( f(-x) ) for ( f(x) = 4x^3 ):

[ f(-x) = 4(-x)^3 = -4x^3 ]

Comparing ( f(-x) ) to ( f(x) = 4x^3 ):

- If ( f(-x) = f(x) ), the function is even. But ( -4x^3 ) is not equal to ( 4x^3 ), so ( f(x) = 4x^3 ) is not even.
- If ( f(-x) = -f(x) ), the function is odd. And since ( -4x^3 = -4x^3 ), the function ( f(x) = 4x^3 ) is odd.

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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.

- How do you find the vertical, horizontal or slant asymptotes for #f(x) = (2x^2 + x + 2) /( x + 1)#?
- How do you identify all asymptotes or holes for #y=(x^3-1)/(x^2+2x)#?
- How do you find the end behavior of #9x^4 - 8x^3 + 4x#?
- What is the range of the function #y = x^2#?
- How do you find the vertical, horizontal or slant asymptotes for # [(9x-4) / (3x+2)] +2#?

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