Answer is: option3

Solution:

A graph is symmetric about the x-axis if:

\( f(x) = -f(x) \)

This means the function would pass the horizontal line test, i.e., for each \( x \), \( f(x) = -f(x) \), which implies:

\[ xe^{-x^2} = -xe^{-x^2} \Rightarrow 2xe^{-x^2} = 0 \Rightarrow x = 0 \]

This is only true at \( x = 0 \), not for all \( x \), so the function is not symmetric about the x-axis.

A graph is symmetric about the y-axis if:

\( f(-x) = f(x) \)

Try plugging in \( -x \):

\[ f(-x) = (-x)e^{-(-x)^2} = -xe^{-x^2} = -f(x) \]

So \( f(-x) = -f(x) \), not equal to \( f(x) \), so it's not symmetric about the y-axis either.

A graph is symmetric about the origin if:

\( f(-x) = -f(x) \)

From above, we found:

\( f(-x) = -f(x) \)

✅ This is true — so the graph is symmetric about the origin.

(C) III only.