Question

In each of Problems 7 through 12 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 \(2 L .\) $$ f(x)=\left\\{\begin{array}{ll}{x,} & {0 \leq x<2} \\ {1,} & {2 \leq x<3}\end{array}\right. $$

Short answer

Please define and find the even and odd extensions of the given function \(f(x)\) on the interval [0, 3), and sketch both extensions on an interval of length 6.

Step-by-step solution

Define even and odd extensions

An even function satisfies \(f(-x)=f(x)\) for all \(x\) in its domain, whereas an odd function satisfies \(f(-x)=-f(x)\) for all \(x\). We can extend a function to be either even or odd by defining it on both sides of the y-axis according to these properties.

Find even and odd extensions of \(f\)

For the given function \(f(x)\) on interval [0, 3), we define its even and odd extensions as follows: Even extension, \(f_e(x)\): $$ f_e(x)=\left\\{ \begin{array}{ll}{x,} & {0 \leq x<2} \\\ {1,} & {2 \leq x<3} \\ {f_e(-x),} & {-3 \leq x<0} \end{array}\right. $$ Odd extension, \(f_o(x)\): $$ f_o(x)=\left\\{ \begin{array}{ll}{x,} & {0 \leq x<2} \\\ {1,} & {2 \leq x<3} \\ {-f_o(-x),} & {-3 \leq x<0} \end{array}\right. $$ Now we need to find \(f_e(x)\) and \(f_o(x)\) for \(-3 \leq x<0\). For even extension, since \(f_e(-x) = f_e(x)\): - if \(-2 \leq x < 0\), then \(f_e(x)=f_e(-x)= -x\) - if \(-3 \leq x <-2\), then \(f_e(x)=f_e(-x)= 1\) For odd extension, since \(f_o(-x) = -f_o(x)\): - if \(-2 \leq x < 0\), then \(f_o(x)=-f_o(-x)= -x\) - if \(-3 \leq x <-2\), then \(f_o(x)=-f_o(-x)= -1\) So the even and odd extensions in interval [-3, 3) are: $$ f_e(x)=\left\\{ \begin{array}{ll}{x,} & {-2 \leq x<2} \\\ {1,} & {-3 \leq x<-2, 2 \leq x<3} \end{array}\right. $$ $$ f_o(x)=\left\\{ \begin{array}{ll}{x,} & {-2 \leq x<2} \\\ {-1,} & {-3 \leq x<-2} \\\ {1,} & {2 \leq x<3} \end{array}\right. $$

Sketch even and odd extensions

Since we need to sketch the extensions for an interval of length \(2L=6\), we just extend our defined range by setting the period \(2L\). Even extension: 1. Start by plotting the line segment from (0, 0) to (2, 2) and from (-2, 2) to (1, -3), representing the interval [-3, 3). 2. Repeat this pattern periodically every 6 units on the x-axis. Odd extension: 1. Start by plotting the line segment from (0, 0) to (2, 2), and from (-2, 2) to (1, -1) representing the interval [-3, 3). 2. Repeat this pattern periodically every 6 units on the x-axis.

Key concepts

Even and Odd Functions

Understanding even and odd functions is an essential part of working with Fourier series. Even functions satisfy the property that their values at positive and negative points are identical. Mathematically, this is expressed as \( f(x) = f(-x) \). This symmetric behavior about the y-axis makes even functions easy to recognize.
Odd functions, on the other hand, have a different type of symmetry. They satisfy the condition \( f(x) = -f(-x) \). This means that an odd function is symmetric about the origin. They will mirror each other in a sort of rotational symmetry as opposed to even functions.
Knowing whether a function is even or odd aids in predicting its behavior across the axis and is fundamental for the analysis using Fourier series, particularly when determining coefficients or simplifying calculations.

Function Extensions

To construct function extensions, you begin by expanding the given function outside its initial domain without altering its original characteristics. Even and odd extensions help in forming periodic functions which are central to Fourier series.
The even extension of a function reflects its values equally across the y-axis, maintaining the even nature through extending like so: if \( f(x) \) for \( x \geq 0 \) is defined, then \( f_e(x) = f(x) \) and \( f_e(-x) = f(x) \) extends it for \( x < 0 \).
Conversely, the odd extension is a bit different. It requires mirroring and negating the values for negative arguments: if \( f(x) \) for \( x \geq 0 \), then \( f_o(x) = f(x) \) and \( f_o(-x) = -f(x) \) extends it for \( x < 0 \).

• Even extension reflects horizontally across the y-axis.
• Odd extension reflects and negates across the origin.

Both techniques ensure that the expanded function fits neatly into an analysis framework, like Fourier series.

Periodic Functions

Periodic functions repeat their values at regular intervals over the x-axis, and this property is central to the Fourier series. When you extend a function to be periodic, you essentially duplicate its behavior over and over again to cover the entire x-axis with consistent intervals.
For instance, if the original interval was from 0 to 3, making it periodic over the interval [0, 6) ensures it repeats every 6 units. This repetition pattern can help simplify both the computation of the series and the analysis of the functions.

• Periodic extensions help apply Fourier series over non-standard intervals.
• Visualizing periodic functions allows for easier prediction of their behavior.
• Conceptual understanding of periodicity aids in resolving complex mathematical phenomena.

This concept helps deal with real-world data where such periodic patterns are common, hence showing the practical relevance of Fourier series.