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.