Consider the following mathematical functions of time ( t ): (I) e^...

A function represents periodic motion if it repeats its values at regular intervals of time, which requires it to be bounded and oscillatory.

(I) $e^{-\omega t}$ decreases monotonically to zero as $t \to \infty$. It does not repeat its values, so it is non-periodic.

(II) $\log_e(\omega t)$ increases monotonically with time. It is non-periodic.

(III) $e^{\omega t}$ increases exponentially with time and is unbounded. It is non-periodic.

(IV) $\sin(\omega t) + \cos(\omega t)$ can be rewritten as $\sqrt{2} \sin(\omega t + \dfrac{\pi}{4})$, which is a bounded, repeating trigonometric function. It represents periodic motion.

Therefore, exactly 3 of the given functions represent non-periodic motion.

Answer: 3