Question

A function \(f\) is given on an interval of length \(L .\) In each case sketch the graphs of the even and odd extensions of \(f\) of period \(2 L .\) $$ f(x)=\left\\{\begin{array}{ll}{0,} & {0 \leq x<1} \\ {1,} & {1 \leq x<2}\end{array}\right. $$

Short answer

Answer: The main differences between the even and odd extensions of the function \(f(x)\) are that the even extension is symmetrical across the y-axis, while the odd extension is symmetrical across the origin.

Step-by-step solution

Sketch the original function f(x)

The first thing to do is sketch the given function on the interval \([0, 2)\). Here's the representation of f(x): $$ f(x) = \left\\{\begin{array}{ll} {0,} & {0 \leq x<1} \\ {1,} & {1 \leq x<2} \end{array}\right. $$ So, the graph starts at \((0,0)\) and is a horizontal line at y=0 until x=1. Then, from x=1 to x=2, the graph is a horizontal line at y=1.

Sketch the even extension of f(x)

To find the even extension of f(x), we will reflect the graph of f(x) across the y-axis, keeping its period equal to \(2L=4\). Following the definition of even functions, reflect f(x) in the interval \([0, 2)\) across the y-axis into the interval \([-2, 0)\). The resulting graph in the interval \([-2, 0)\) will be the same as the original graph in interval \([0,2)\) but flipped, so that the left-side graph mirrors the right side graph across the y-axis. Repeat the process every 4 units (the period of the even extension) to sketch the full even extension of f(x).

Sketch the odd extension of f(x)

To find the odd extension of f(x), we will reflect the graph of f(x) across the origin, keeping its period equal to \(2L=4\). Following the definition of odd functions, reflect f(x) in the interval \([0, 2)\) across the origin into the interval \([-2, 0)\). The resulting graph in the interval \([-2, 0)\) will be the original graph in interval \([0,2)\) but flipped and rotated 180 degrees, meaning it starts at \((0,0)\), has the value 1 in the interval \((-2,-1)\), and is 0 in the interval \((-1,0)\). Repeat the process every 4 units (the period of the odd extension) to sketch the full odd extension of f(x). Now that we have identified the even and odd extensions for the function \(f(x)\) of period \(2L=4\), we can see the main differences between them; the even extension is symmetrical across the y-axis, while the odd extension is symmetrical across the origin.

Key concepts

Understanding Differential Equations in the Context of Function Extensions

Differential equations are mathematical equations that involve an unknown function and its derivatives. They play a crucial role in describing various physical and theoretical systems where change is a fundamental aspect. When dealing with differential equations, particularly in the context of periodic functions and their extensions, it's important to consider the symmetry properties of these functions.

Even and odd extensions of functions are pertinent when solving certain boundary value problems involving differential equations. These extensions ensure that the solutions are consistent with the physical or theoretical constraints of the problem, such as the continuity and differentiability over the entire domain. In the example of the step function provided, the even extension preserves the values of the function across the y-axis; hence, its derivative (if it exists) will be symmetric as well. An odd extension, conversely, results in a function whose values are mirrored and flipped about the origin, which impacts the sign of the derivative when taken across the interval. This concept is fundamental when considering the periodicity and symmetrical properties required by solutions to differential equations.

Boundary Value Problems and Their Relation to Function Extensions

Boundary value problems are a type of differential equation where the solution is sought within a certain interval and must satisfy specific conditions at the boundaries. These problems are particularly challenging because the solutions must be consistent at the endpoints, which is where even and odd functions come into play.

The extension of a piecewise function as even or odd ensures that the boundary conditions are met. For even extensions, the function values at symmetric points relative to the y-axis are equal, thus satisfying the boundary conditions at those points. Meanwhile, odd extensions have symmetric points with equal magnitude but opposite signs, often required in systems that are antisymmetric about the origin.

In practical scenarios, such as physics and engineering, boundary value problems arise in the study of steady-state solutions to heat equations, vibration modes of a string, and electrostatic potential distributions, where the type of function extension directly affects the behavior and the validity of the solutions.

Graphing Piecewise Functions and Their Symmetric Extensions

Graphing piecewise functions involves drawing functions that have different definitions over different intervals. These functions often emerge in real-world situations where a system behaves differently under various conditions. When extending piecewise functions to cover a larger domain, choosing an appropriate extension - either even or odd - becomes crucial, depending on the context.

An even extension of a piecewise function involves mirroring the function about the y-axis, creating a symmetric graph that repeats every period. In contrast, an odd extension involves rotating the function 180 degrees about the origin, resulting in a graph that is antisymmetric with a similar periodic repetition.

Visualizing the Extensions

When sketching these extensions, it's essential to accurately reflect the behavior of the function in its original interval across the appropriate axes. This visualization can aid in understanding the nature of the solution when used in boundary value problems and differential equations. The step function from the given exercise, when extended, exhibits these properties clearly, providing a straightforward example of how different types of function extensions impact the overall graph of a solution.