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

• 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.

#color(white)()#
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.

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.