Problem 8 - Applied Partial Differential Equations

Question

(a) Determine formulas for the even extension of any \(f(x)\). Compare to the formula for the even part of \(f(x)\). (b) Do the same for the odd extension of \(f(x)\) and the odd part of \(f(x)\). (c) Calculate and sketch the four functions of parts (a) and (b) if $$ f(x)= \begin{cases}x & x>0 \\ x^{2} & x<0\end{cases} $$ Graphically add the even and odd parts of \(f(x) .\) What occurs? Similarly, add the even and odd extensions. What occurs then?

Step-by-Step Solution

Answer

The formula for even extension is \[E_f(x) = \frac{1}{2}(f(x)+f(-x))\] and for odd extension is \[O_f(x) = \frac{1}{2}(f(x)-f(-x))\]. When we apply these to the given function and sum the even and odd parts of \(f(x)\), we get the original \(f(x)\). Similarly, sum of even and odd extensions resembles the reflection of original function about y-axis.

Step 1: Define Even Extension

An even function satisfies the property \(f(-x) = f(x)\) for all \(x\) in the domain. The even extension of \(f(x)\) is defined as \[E_f(x) = \frac{1}{2}(f(x)+f(-x)).\]

Step 2: Define Odd Extension

An odd function satisfies the property \(f(-x) = -f(x)\) for all \(x\) in the domain. The odd extension of \(f(x)\) is defined as \[O_f(x) = \frac{1}{2}(f(x)-f(-x)).\]

Step 3: Analyze Even and Odd parts of function

The even part of \(f(x)\) is when \(x\) is positive, the odd part of \(f(x)\) is when \(x\) is negative.

Step 4: Apply formulas to given function

The given function is \(f(x)=x\) for \(X>0\) and \(f(x)=x²\) for \(x<0\). The even extension of this function would be \[E_f(x) = \frac{1}{2}(f(x)+f(-x))\] for \(x>0\) and \[E_f(x) = \frac{1}{2}(f(x)+f(-x))\] for \(x<0\). The odd extension would be \[O_f(x) = \frac{1}{2}(f(x)-f(-x))\] for \(x>0\) and \[O_f(x) = \frac{1}{2}(f(x)-f(-x))\] for \(x<0\).

Step 5: Sketch Graphs

A graphical representation of these equations will provide visual clarity. For the even extension, for \(x>0\) we have a linear function while for \(x<0\) we have a quadratic function. For the odd extension, we have the same but the direction is opposite.

Step 6: Analyzing addition of functions

Adding the even and odd parts of \(f(x)\) will basically present the original function. For the even extension, it will be symmetric about the y-axis. Similar case for the odd extension but it should be antisymmetric.

Key Concepts

Even Function

An even function is one that satisfies the condition \(f(-x) = f(x)\) for all values of \(x\) within its domain. This means that the graph of an even function is symmetrical with respect to the y-axis. If you fold the graph along the y-axis, both halves will match perfectly.

To understand even extensions better, imagine you are given a function \(f(x)\). By constructing the even extension \(E_f(x)\), we find a new function that reflects \(f(x)\) symmetrically over the y-axis:

• For positive \(x\), it replicates the original function \(f(x)\).
• For negative \(x\), it mirrors the value at the positive side, \(f(-x) = f(x)\).

The formula for creating this even extension is:\[E_f(x) = \frac{1}{2}(f(x) + f(-x))\]This calculation averages \(f(x)\) and \(f(-x)\), ensuring symmetry across the y-axis.

Odd Function

An odd function satisfies the property that \(f(-x) = -f(x)\) for every \(x\) in its domain. The graph of an odd function is symmetrical around the origin. Visually, this means that if you rotate the graph by 180 degrees around the origin, it will look the same.

To extend a given function into an odd function using its odd extension, you follow a process that creates a new function with a center of symmetry at the origin:

• For positive \(x\), the function remains as \(f(x)\).
• For negative \(x\), it transforms into \(-f(x)\).

The expression that captures this odd extension is:\[O_f(x) = \frac{1}{2}(f(x) - f(-x))\]This ensures that each pair of values at \(x\) and \(-x\) are opposite, preserving the symmetry about the origin.

Piecewise Function

A piecewise function is a function that is defined by different expressions for different parts of its domain. This approach is ideal when different rules or behaviors are necessary for various intervals of input values.

For example, in our exercise, the function \(f(x)\) is specified as:

• \(f(x) = x\) when \(x > 0\)
• \(f(x) = x^2\) when \(x < 0\)

This means that when you analyze the function or extend it—for even or odd parts—you deal with each section separately. Essentially, you calculate the even or odd extensions based on the specific expression that is valid in each segment of \(x\).

Understanding the behavior of piecewise functions is crucial, particularly because you might encounter scenarios where displaying, extending, or analyzing only one part at a time is necessary.

Graphical Analysis

Graphical analysis is a valuable tool for understanding the behavior and properties of functions, especially when dealing with even and odd functions or extensions. By sketching the graph, one can visually confirm the symmetry properties inherent in even and odd functions.

For the function in our exercise:

• We first draw the original piecewise function, recognizing the linear and quadratic forms for \(x > 0\) and \(x < 0\) respectively.
• The even extension appears symmetrical about the y-axis. If you plot \(E_f(x)\), you'll see how each point is "mirrored" over the y-axis.
• The odd extension shows symmetrically around the origin, where each positive point at \(x\) has a corresponding negative counterpart at \(-x\).

Furthermore, observing the graphs of both the even and odd extensions can greatly aid in understanding how the function behaves both separately and together with its even and odd parts, creating a comprehensive picture of the function's properties.