Final answer:
The product of two periodic functions is periodic when the ratio of their periods is a rational number, allowing them to recur at the least common multiple (LCM) of the periods.
Step-by-step explanation:
The product of two periodic functions is periodic if the ratio of their periods is a rational number. The product of two periodic functions is again periodic when the ratio of their frequencies is a rational number. In other words, if the ratio of the two periods is a rational number, then the product of the functions will be periodic.
In mathematics, periodic functions have a regular repeating pattern. To be periodic, the product of two functions f(x) and g(x) must repeat its values at constant intervals, called periods. If f(x) has a period T1 and g(x) has a period T2, then for the product to be periodic, the ratio T1/T2 must be a rational number, which can be expressed as a fraction p/q where p and q are integers. This ensures that the product f(x)*g(x) recurs at regular intervals, specifically at the least common multiple (LCM) of the two periods.
Considering frequency and period in oscillations, frequency f is the number of oscillations per unit time and is the inverse of the period T. The two quantities are inversely related, as expressed by the equation f = 1/T.