Determine whether the following function is even, odd, or neither even nor odd. \( f(x)=2 x \sqrt{10-x^{2}} \) Choose the correct answer below. A. The function \( f \) is even. B. The function \( f \) is odd. C. The function \( f \) is neither even nor odd.

To determine the nature of the function \( f(x) = 2x \sqrt{10 - x^2} \), we need to evaluate \( f(-x) \): \[ f(-x) = 2(-x) \sqrt{10 - (-x)^2} = -2x \sqrt{10 - x^2} = -f(x). \] Since \( f(-x) = -f(x) \), this means that the function \( f \) is odd. So the correct answer is: B. The function \( f \) is odd. Now, here’s something fun about odd functions: they are typically symmetric about the origin! This means that if you were to take the graph of an odd function and rotate it 180 degrees around the origin, it would look exactly the same. For real-world applications, odd functions can be found frequently in physics, especially in situations involving forces and momentum, as they often represent changes in direction with respect to a variable. For example, an odd function can depict the restoring force in certain pendulum systems, reflecting how the force works in opposite directions based on displacement.