Problem 75 - Thomas Calculus

Question

Show that if a function \(f\) is defined on an interval symmetric about the origin \((\) so that \(f\) is defined at \(-x\) whenever it is defined at \(x)\) then $$f(x)=\frac{f(x)+f(-x)}{2}+\frac{f(x)-f(-x)}{2}$$ Then show that \((f(x)+f(-x)) / 2\) is even and that \((f(x)-\) \(f(-x) ) / 2\) is odd.

Step-by-Step Solution

Answer

The function decomposes into even and odd parts, satisfying the given expression.

Step 1: Expand the Function Expression

We start with the expression given in the problem:\[ f(x) = \frac{f(x) + f(-x)}{2} + \frac{f(x) - f(-x)}{2} \]This expression splits the function into two parts, the first involves an average of \( f(x) \) and \( f(-x) \), and the second involves their difference, still averaged.

Step 2: Identify the Even Function Component

The expression \( \frac{f(x) + f(-x)}{2} \) is symmetric about the y-axis. To show this component is an even function, check its definition under \(-x\):\[ \frac{f(-x) + f(x)}{2} = \frac{f(x) + f(-x)}{2} \]This confirms the component is unchanged by negation, satisfying the condition for evenness, \(g(x) = g(-x)\).

Step 3: Identify the Odd Function Component

Now consider the expression \( \frac{f(x) - f(-x)}{2} \). Check for symmetry about the origin:\[ h(-x) = \frac{f(-x) - f(x)}{2} = -\frac{f(x) - f(-x)}{2} \]This changes sign with \(x\), satisfying the condition for oddness, \(h(-x) = -h(x)\).

Step 4: Verify Identity of Function

Adding the results from Steps 2 and 3, observe:\[ g(x) + h(x) = \frac{f(x) + f(-x)}{2} + \frac{f(x) - f(-x)}{2} = f(x) \]Hence, or given a function both terms together reconstruct the original function \( f(x) \).

Key Concepts

Function Symmetry

Understanding function symmetry is crucial in mathematics. Symmetry refers to a function's behavior when plotted on a graph. Function symmetry is where a function mirrors itself across a certain line or point. Here, the function is defined on an interval symmetric about the origin, meaning if the function is defined at a point \(x\), it should also be defined at \(-x\). This concept helps in splitting functions into parts that can be more easily analyzed. By examining both the even and odd components separately, complex functions become much simpler to handle and understand.

Mathematical Proof

Mathematical proofs are structured logical arguments that verify the truths of mathematical statements. In this exercise, a proof verifies that a function can be expressed as a sum of even and odd components. The given equation is:\[f(x)=\frac{f(x)+f(-x)}{2}+\frac{f(x)-f(-x)}{2}\]Each part of this equation has a role in expressing the function's symmetry. By showing:

• \(\frac{f(x) + f(-x)}{2}\) is unchanged under the operation of negation, i.e., it is even.
• \(\frac{f(x) - f(-x)}{2}\) changes sign under negation, i.e., it is odd.

Adding these components reconstructs the original function, confirming the identity. This process underlines how mathematical proofs confirm theories and properties using logical deduction.

Functional Decomposition

Functional decomposition involves breaking down complex functions into simpler components. It's like peeling an orange into slices: each slice represents a simpler component of the whole. In the exercise, the function \(f(x)\) is decomposed as:

• An even component: \(\frac{f(x) + f(-x)}{2}\)
• An odd component: \(\frac{f(x) - f(-x)}{2}\)

This separation simplifies analysis since each component displays distinct symmetry properties: evenness and oddness. Functional decomposition aids in understanding and analyzing various features of functions such as behavior, roots, and graph symmetry. By simplifying a function into even and odd parts, you gain clearer insights into its nature.

Symmetric Interval

A symmetric interval in mathematics refers to an interval that spans equally in both directions from a central point, often the origin. For example, an interval \([-a, a]\) is symmetric about zero. This exercise uses a symmetric interval where the function is defined such that if \(x\) is included, then \(-x\) must be included as well.

• This property of symmetry ensures that all operations, like taking averages or differences, can be performed cohesively.
• The symmetric interval allows functions to be evaluated for their even and odd parts consistently.

Understanding symmetric intervals is fundamental in analyzing functions because it sets the stage for applying even-odd decomposition and helps identify algebraic properties related to symmetry.