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)=x-3, \quad 0

Short answer

Answer: The even extension of the function in the interval (-4, 4) is: $$ f_e(x) = \begin{cases} x-3, & 0<x<4 \\ -x-3, & -4<x<0 \end{cases} $$ And the odd extension of the function in the interval (-4, 4) is: $$ f_o(x) = \begin{cases} x-3, & 0<x<4 \\ 3-x, & -4<x<0 \end{cases} $$

Step-by-step solution

Even Extension

To find the even extension of \(f(x)\) with period \(8\), we need to make sure the function satisfies \(f(x)=f(-x)\) for all \(x\). First, let's express the current function within the given interval: $$ f(x) = \begin{cases} x-3, & 04} \end{_cases} $$ Now, let's extend it to satisfy the condition \(f(x) = f(-x)\): $$ \text{Even Extension}(x)=f_e(x) = \begin{cases} x-3, & 0<x<4 \\ -x-3, & -4<x<0 \\ \left(\text{repeat for other periods of 8}\right) \end{_cases} $$ To sketch the graph, plot the function \(f(x)=x-3\) for \(0<x<4\) and \(f(x)=-x-3\) for \(-4<x<0\), and keep repeating the resulting graph for every interval of length \(8\).

Odd Extension

To find the odd extension of \(f(x)\) with period \(8\), we need to make sure the function satisfies \(f(x)=-f(-x)\) for all \(x\). First, let's express the current function within the given interval: $$ f(x) = \begin{cases} x-3, & 04 \end{cases} $$ Now, let's extend it to satisfy the condition \(f(x) = -f(-x)\): $$ \text{Odd Extension}(x)=f_o(x) = \begin{cases} x-3, & 0<x<4 \\ 3-x, & -4<x<0 \\ \left(\text{repeat for other periods of 8}\right) \end{cases} $$ To sketch the graph, plot the function \(f(x)=x-3\) for \(0<x<4\) and \(f(x)=3-x\) for \(-4<x<0\), and keep repeating the resulting graph for every interval of length \(8\).

Key concepts

Even Extension

An even extension of a function is a way to "mirror" the function across the y-axis. For a function like our example, which is only defined between 0 and 4, extending it to be an even function means we have to define it from -4 to 4 in such a way that it remains symmetric around the y-axis.
This symmetry means that if you take a point on the positive half of the x-axis, the function value at this point should be the same as the function value at the corresponding negative point. Mathematically, this is expressed as \(f(x) = f(-x)\).
To create this extension, you simply "flip" the part of the function that is already defined to the opposite side of the y-axis. In this case:

• For \(x\) from 0 to 4, the function is \(f(x) = x - 3\), straightforward as is.
• For \(x\) from -4 to 0, the "flipped" function will be \(f(x) = -x - 3\), making an even continuation for negative x-values.

Extend this pattern to achieve periodicity with any specified period such as 8 for our function.
It's a simple yet powerful concept, helping us understand symmetrical functions and prepare for Fourier analysis.

Odd Extension

Odd extensions are the counterpart to even extensions. Instead of mirroring, we "rotate" the function around the origin. In mathematical terms, odd extensions satisfy \(f(x) = -f(-x)\).
This condition means that when you take a point at \(x\), its corresponding function value in the negative side \(-x\) should be the negative of the original value. Basically, if you know one half, you use the negative to find the other.
Let's break this down with our example:

• From 0 to 4, the original function is still \(f(x) = x - 3\).
• From -4 to 0, the function is extended as \(f(x) = 3 - x\), maintaining the odd symmetry.

This doesn't stop here. For a truly periodic function, this pattern has to repeat for each new period (such as every 8 units on the x-axis).
Odd extensions are crucial for constructing Fourier series, especially when analyzing signals or vibrations."

Periodic Functions

A function is called periodic if it repeats itself at regular intervals. This interval, known as the period, is a constant value, and the function looks identical after each interval.
In formulas, a function \(f(x)\) is periodic if there exists a positive constant \(T\) such that \(f(x + T) = f(x)\) for every \(x\).
Understanding periodic functions is essential, as they are everywhere in nature and engineering, especially in waveforms and signals.

• The period for the given function \(f(x) = x - 3\) is doubled by the extension process, resulting in a new period of 8.
• Within this period, both the even and odd extensions allow the function to perfectly repeat itself.

Periodic functions are central to Fourier analysis, because Fourier series represent any periodic function as a sum of sine and cosine waves. This is incredibly useful for signal processing, acoustics, and many scientific fields.
By mastering periodic functions, you pave the way for understanding complex wave patterns and more!