Question

Determine whether the given function is periodic. If so, find its fundamental period. $$ f(x)=\left\\{\begin{array}{ll}{0,} & {2 n-1 \leq x<2 n,} \\ {1,} & {2 n \leq x<2 n+1 ;}\end{array} \quad n=0, \pm 1, \pm 2, \dots\right. $$

Short answer

If yes, what is its fundamental period? Answer: Yes, the given function is periodic with a fundamental period of T = 4.

Step-by-step solution

Check if the function is periodic

To check if the function is periodic, we need to find a value for the period T such that f(x + T) = f(x) for all x. Let's compare the function values at different intervals of x: - Compare f(x) and f(x+2) when x lies in the interval [2n-1, 2n) for any n: f(x) = 0 in this interval, and (x+2) lies in the interval [2n+1, 2n+2) for the same value of n. In this interval, f(x+2) = f([2n+1, 2n+2)) = 1, So, f(x) ≠ f(x+2) in this interval. - Compare f(x) and f(x+4) when x lies in the interval [2n-1, 2n) for any n: f(x) = 0 in this interval, and (x+4) lies in the interval [2n+3, 2n+4) for the same value of n. In this interval, f(x+4) = f([2n+3, 2n+4)) = 0, So, f(x) = f(x + 4) in this interval. We observe that the function values repeat after every 4 units. Therefore, the function is periodic.

Determine the fundamental period

Since f(x) = f(x + 4) for every x, it means the function's values repeat after every 4 units. Therefore, the fundamental period of the given function is T = 4. So, the given function is periodic with a fundamental period of T = 4.

Key concepts

Fundamental Period

When exploring the concept of a fundamental period, we delve into the heartbeat of periodic functions. By definition, a function is said to be periodic if there exists a positive number, often denoted as T, for which the property f(x + T) = f(x) holds true for all values of x. The smallest such value of T is known as the fundamental period. It's a measure of how frequently the function repeats itself, effectively setting the rhythm for its oscillations.

In the case of our function f(x), which is defined piecewise for intervals of the form [2n-1, 2n) and [2n, 2n+1), we determined the fundamental period to be 4. This means that after every 4 units along the x-axis, the function's values create an identical pattern, akin to a well-timed dance, continuing ad infinitum.

Piecewise Functions

Diving into the realm of piecewise functions, we recognize a function that is defined by different expressions over various intervals. Think of it as a quilt made up of different fabric patterns, with each piece having its own unique design. In mathematical terms, the function may take on several forms depending on where you are within its domain.

For our textbook exercise, the function f(x) is defined in two alternating pieces—one for even and another for odd intervals of the independent variable, marked by n, which is an integer. Understanding piecewise functions is crucial because real-world phenomena often change behavior abruptly rather than following a single regular rule, mirroring how our function behaves differently on adjacent intervals.

Periodic Function Properties

The characteristics of periodic function properties are essential in unraveling the function's behavior. Periodicity is not just about repetition; it pertains to symmetry, regularity, and even predictability. For a function to be periodic, it must satisfy certain criteria beyond the simple repetition of values. For instance, both the start and end points of one period must connect seamlessly to the corresponding points in adjacent periods.

Moreover, the function's properties must be consistent across each period. This includes not only the function's values but also its rate of change, curvature, and other differential properties. In our function f(x), it shows a perfect jump between 0 and 1 as its defining property that recurs every 4 units, embodying the conceptual purity of periodic function properties.

Differential Equations

Differential equations are equations that involve an unknown function and its derivatives. They are a cornerstone of understanding how quantities change over time or space and are profoundly rooted in both natural laws and mathematical theory. Although our textbook function is not explicitly defined by a differential equation, understanding such equations is invaluable for analyzing a vast array of periodic phenomena.

When dealing with periodic behavior, solutions to differential equations can often denote harmonic motion or oscillations represented by sine and cosine functions, which themselves are quintessential periodic functions. Recognizing the connection between differential equations and periodic functions provides context and insight when dissecting the periodic elements of complex systems in fields spanning from physics to finance.