Question

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

Answer 1

`f(x) = 4x^3` is an odd function.

Explanation:

• An even function is one for which `f(-x) = f(x)` for all `x` in its domain.

• An odd function is one for which `f(-x) = -f(x)` for all `x` in its domain.

In our example:

`f(-x) = 4(-x)^3 = -4x^3 = -f(x)`

for all values of `x`.

So `f(x) = 4x^3` is an odd function.

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Footnote

For polynomials, there is a shortcut to telling whether it is odd or even:

Are all of the terms of odd degree, even degree or a mixture?

If odd then the function is odd. If even then the function is even. If neither then it is neither.

Note that constant terms are of even (`0`) degree.

Answer 2

odd function

Explanation:

To determine if a function f(x) is even/odd consider the following.

• If f(x) = f( -x) , then f(x) is even

Even functions are symmetrical about the y-axis.

• If - f(x) = f(-x) , then f(x) is odd

Odd functions have symmetry about the origin.

Test for even function

`f(-x)=4(-x)^3=-4x^3≠f(x)`

Since f(x) ≠ f( -x) , then f(x) is not even

Test for odd function

`-f(x)=-(4x^3)=-4x^3=f(-x)`

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

Note symmetry about the origin.