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)=2-x, \quad 0

Short answer

In conclusion, the even extension of the function \(f(x) = 2 - x\) with a period of \(4\) is given by \(g_{even}(x + 4n) = 2 - |x + 4n|\), while the odd extension is given by \(g_{odd}(x + 4n) = sgn(x + 4n) \cdot (2 - |x + 4n|)\), for all integers \(n\). By sketching the original function, the even and odd extensions, and their periodicity, we can visualize the required solutions for this problem.

Step-by-step solution

Define the Even and Odd Extensions

An even extension of a function \(f(x)\) is defined as \(g(x) = f(|x|)\). An odd extension of the function \(f(x)\) is defined as \(g(x) = sgn(x) \cdot f(|x|)\), where \(sgn(x)\) is the sign function, which returns \(1\) if \(x > 0\), \(-1\) if \(x < 0\), and \(0\) if \(x = 0\).

Find Even and Odd Extensions for Given Function

We are given the function \(f(x) = 2-x\) for \(0 < x < 2\). For the even extension, we get $$ g_{even}(x) = f(|x|) = 2 - |x|, $$ and for the odd extension, we get $$ g_{odd}(x) = sgn(x) \cdot f(|x|) = sgn(x) \cdot (2 - |x|). $$

Find the Periodic Extensions of the Functions

Now we need to find the periodic extensions of \(g_{even}(x)\) and \(g_{odd}(x)\) with a period of \(4\). For the even extension, the periodic extension is given by $$ g_{even}(x + 4n) = 2 - |x + 4n|, $$ for all integers \(n\). For the odd extension, the periodic extension is given by $$ g_{odd}(x + 4n) = sgn(x + 4n) \cdot (2 - |x + 4n|), $$ for all integers \(n\).

Sketch the Graphs of the Extensions

To sketch the graphs of the even and odd extensions, you can follow these steps: 1. Sketch the original function \(f(x) = 2 - x\) over the interval \((0, 2)\). 2. For the even extension \(g_{even}(x) = 2 - |x|\), reflect the original graph along the y-axis and draw its periodic extensions in both positive and negative directions with a period of \(4\). 3. For the odd extension \(g_{odd}(x) = sgn(x) \cdot (2 - |x|)\), reflect the original graph along the y-axis with a negative sign, and draw its periodic extensions in both positive and negative directions with a period of \(4\). The provided sketches of the even and odd extensions will represent the required solutions for the given exercise.

Key concepts

Periodic Functions

Periodic functions are mathematical functions that repeat their values in regular intervals or periods. Imagine the hands of a clock that return to the same position every 12 hours — they exhibit periodicity. Mathematically, if a function has period L, for all values of x in the function's domain, f(x + L) = f(x). This is the essence of a periodic function.

The exercise presents a simplistic example: after extending a function evenly or oddly, you often transform it into a periodic function with a specific period 2L. The periodic nature becomes apparent when sketching the graph, as the repeating pattern emerges. Recognizing and drawing these patterns are crucial in understanding the behavior of trigonometric functions like sine and cosine, which are fundamental in physics, engineering, and signal processing.

Extension of a Function

An extension of a function is a concept where the original function defined on a specific interval is expanded beyond that interval, while maintaining certain properties. In the given exercise, the function f(x) = 2 - x is originally defined on the interval (0,2). To extend it and preserve particular characteristics, we create an even and an odd extension. The even extension of a function is symmetrical about the y-axis, while the odd extension is symmetrical about the origin. These types of extensions allow for the analysis and further application of the function; for instance, in Fourier series where even and odd functions play a crucial role in simplifying computations. By extending functions, mathematicians and engineers can solve a variety of real-world problems — from acoustic engineering to image processing.

Sketching Graphs of Functions

Sketching graphs of functions is a vital skill in mathematics. It transforms abstract equations into a visual format, providing a clear understanding of the function's behavior. The step-by-step solution for the exercise involved sketching the even and odd extensions. To do this proficiently, start with the original function within its interval. Then, use symmetry for the even extension, or reflection and alternation of signs for the odd extension, to continue the graph beyond the original interval.

When sketching, keep in mind the periodicity and ensure that these properties are consistently applied across the entire set of periods. Graphing calculators or software can be extremely beneficial for visualizing these concepts, especially when dealing with more complex or abstract functions.

Sign Function

The sign function, often denoted as sgn(x), is a simple, yet profound mathematical function. It extracts the sign of a real number, assigning +1 to positive numbers, -1 to negative numbers, and 0 to zero itself. This function is instrumental in creating odd extensions of a given function.

In the context of the exercise, the sign function modifies the even extension of the original function f(x) to create an odd function. By multiplying the even extension by sgn(x), the resulting graph will be symmetrical along the origin, changing sign as x crosses from positive to negative, thus exhibiting the characteristics of an odd function. Understanding the sign function is essential when analyzing waveforms, designing control systems, and studying mathematical series.