Is the function #f(x) = x^3# symmetric with respect to the y-axis?

1 Answer
Mar 18, 2018

Answer:

No, it has rotational symmetry of order #2# about the origin.

Explanation:

  • An even function is a function satisfying:
    #f(-x) = f(x)" "# for all #x# in the domain of #f(x)color(white)(0/0)#

  • An odd function is a function satisfying:
    #f(-x) = -f(x)" "# for all #x# in the domain of #f(x)color(white)(0/0)#

Even functions are symmetric with respect to the #y#-axis.

Odd functions have rotational symmetry of order #2# about the origin.

Given:

#f(x) = x^3#

Note that for any value of #x#:

#f(-x) = (-x)^3 = (-1)^3 x^3 = -x^3 = -f(x)#

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

It is not symmetric with respect to the #y#-axis, but it has rotational symmetry of order #2# about the origin.

graph{x^3 [-5, 5, -10, 10]}

In fact any polynomial consisting of only terms of odd degree will be an odd function and any polynomial consisting of only terms of even degree will be an even function.