Question

Square wave Graph the square wave defined by $$f(x)=\left\\{\begin{array}{ll}0 & \text { if } x<0 \\\1 & \text { if } 0 \leq x<1 \\\0 & \text { if } 1 \leq x<2 \\\1 & \text { if } 2 \leq x<3\end{array}\right.$$

Short answer

Answer: The square wave function has a repeating pattern of having a value of 1 for the interval [n, n+1) and a value of 0 for the interval [n+1, n+2), where n is any even integer. On the graph, this pattern is represented by alternating horizontal lines parallel to the x-axis at y = 1 and y = 0, with open circles at points where the function is not defined.

Step-by-step solution

Analyze the different definitions of f(x)

In order to graph the square wave f(x), we need to understand each piece of the function definition. The function is defined as: - 0, if x < 0 - 1, if 0 ≤ x < 1 - 0, if 1 ≤ x < 2 - 1, if 2 ≤ x < 3

Determine the limits of the function on each interval

Now that we understand the definitions, determine the function values and mark the limits on the graph. Based on the function definition, we can determine the values for f(x) at the limits of the intervals: - For x < 0, the function is 0. - At x = 0, the function is 1. - At x = 1, the function is 0. - At x = 2, the function is 1. - At x = 3, the function is not defined.

Graph the square wave for the given intervals

Now, we will graph the function for each interval: - For x < 0: f(x) is 0, so it is a horizontal line along the x-axis. - For 0 ≤ x < 1: The function is 1, so it is a horizontal line parallel to the x-axis at y = 1 for this interval. - For 1 ≤ x < 2: The function is 0, so it is a horizontal line along the x-axis, touching the previous interval. - For 2 ≤ x < 3: The function is 1, so it is a horizontal line parallel to the x-axis at y = 1 for this interval. When sketching the graph, indicate open circles at points where the function is not defined on the intervals.

Observe and extend the pattern

From the graph, we can see that the square wave function has a repeating pattern of having value 1 for the interval [n, n+1) and value 0 for the interval [n+1, n+2) where n is any even integer. To graph the function for a larger range, continue this pattern to the left and to the right.

Key concepts

Square Wave Function

The square wave function is a fascinating and rich mathematical concept that represents a periodic waveform with distinct values over specific intervals. Think of it like a toggle switch flipping on and off at set points along the x-axis on a graph.

In our case, the square wave function, denoted as \( f(x) \), jumps between the values of 0 and 1. When graphing such a function, you'll notice it produces a sequence of squares or rectangles – hence the name 'square wave'. One of the most intriguing aspects of the square wave is its non-smooth, abrupt changes, which make it a staple example when discussing discontinuities in functions.

This kind of waveform is not just an abstract notion; it plays a critical role in electronics and signal processing. For example, square waves are a basic building block in digital circuits and synthesizers, making them fundamental in producing sounds in electronic instruments.

Piecewise Function Definition

A piecewise function is like a multipart story, where each part of the narrative has its own rules. In mathematics, this translates to a function that is defined by different expressions based on the interval of the input value. These individual expressions are like chapters of a book, each describing a segment of the function's overall behavior.

The formal definition goes something like this: a piecewise function is a function that is described by multiple sub-functions, each of which applies to a certain interval of the domain (all the possible inputs). When graphed, a piecewise function may resemble a mosaic, where each piece fits perfectly within its own designated boundaries but may differ in appearance and value from adjacent pieces.

Learning to craft a piecewise function requires both analytical skills to understand different conditions and graphical skills to represent these conditions intuitively on a coordinate plane.

Function Intervals

Where a story is divided into chapters, piecewise functions are divided into function intervals. Each interval denotes a specific range of input values for which the function follows a particular rule. For example, imagine you're setting conditions for a game—the rules of the game change depending on the intervals of time that have passed.

In the given exercise, the square wave function is defined by intervals on the x-axis such as from \( 0 \) to \( 1 \), \( 1 \) to \( 2 \), and so on. The function exhibits a different behavior or output within each of these intervals, making it critical to precisely determine these ranges.

The interval notation, such as \( [0,1) \), describes a range where the function operates under a specific definition. Square brackets signify inclusion, while parentheses denote exclusion. Thus, in the interval \( [0,1) \), the value at \( x = 0 \) is included in the range, whereas the value at \( x = 1 \) is not.

Continuity of Functions

The smoothness of a road or path often determines the comfort of a journey. Similarly, in mathematics, the concept of continuity of functions deals with how smoothly a graphed function transitions from one point to the next without breaks, jumps, or abrupt changes in direction.

A function is considered continuous at a point if you can draw its curve at that point without lifting your pencil off the paper. This means that the function's value at that point is well-defined, and it matches both the incoming and the outgoing limits.

The square wave function, however, serves as a classic example of discontinuity in certain places. For instance, it's not continuous at the points where it jumps from 0 to 1 or from 1 to 0. In mathematical terms, the left-hand limit and the right-hand limit do not match at these points, and hence, you'll have to 'lift your pencil off the paper' to continue drawing the function. It's this fascinating property of continuity (or the lack thereof) that encapsulates both the beauty and complexity of piecewise functions.