Question

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

Answer 1

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.