Answer
The function is odd, meaning it is symmetric with respect to the origin.
Work Step by Step
Even functions are symmetric with respect to the y-axis, and odd functions are symmetric with respect to the origin.
To test for symmetry with respect to the x-axis, we substitute g(x) for g(x) and check if it equals the original equation:
$-g(x)=\dfrac{2x^3}{x^4+1}$
$g(x)=-\dfrac{2x^3}{x^4+1}$ nope
To test for symmetry with respect to the y-axis, we substitute x for -x and check if it equals the original equation:
$g(x)=\dfrac{2(-x)^3}{(-x)^4+1}$
$g(x)=\dfrac{-2x^3}{x^4+1}$ nope
To test for symmetry with respect to the origin, we substitute x for -x, substitute g(x) for -g(x) and check if it equals the original equation:
$-g(x)=\dfrac{2(-x)^3}{(-x)^4+1}$
$g(x)=-\dfrac{-2x^3}{x^4+1}$
$g(x)=\dfrac{2x^3}{x^4+1}\checkmark$
