Problem 23 - Calculus Early Transcendentals: Pearson New International Edition

Question

Let \(F\) be any function whose domain contains \(-x\) whenever it contains \(x\). Prove each of the following. (a) \(F(x)-F(-x)\) is an odd function. (b) \(F(x)+F(-x)\) is an even function. (c) \(F\) can always be expressed as the sum of an odd and an even function.

Step-by-Step Solution

Answer

(a) Odd, (b) Even, (c) Possible as the sum of an odd and even function.

Step 1: Understand What is Given

We are given a function \(F\) that, whenever it includes \(x\) in the domain, must also include \(-x\). We are to prove certain properties about its combinations and decompositions.

Step 2: Prove (a) - Odd Function Property

We need to show that \(F(x) - F(-x)\) is an odd function. A function \(f(x)\) is odd if \(f(-x) = -f(x)\) for all \(x\) in its domain. Consider \(f(x) = F(x) - F(-x)\).Now, compute \(f(-x):\) \[ f(-x) = F(-x) - F(x)\] Notice that this is indeed equal to \[-(F(x) - F(-x)) = -f(x)\]Hence, \(F(x) - F(-x)\) is odd.

Step 3: Prove (b) - Even Function Property

Next, we prove \(F(x) + F(-x)\) is an even function. A function \(g(x)\) is even if \(g(-x) = g(x)\). Consider \(g(x) = F(x) + F(-x)\).Now compute \(g(-x):\) \[g(-x) = F(-x) + F(-(-x)) = F(-x) + F(x)\] Since addition is commutative, we have \[F(-x) + F(x) = F(x) + F(-x) = g(x)\]Hence, \(F(x) + F(-x)\) is even.

Step 4: Prove (c) - Expression as Sum of Odd and Even Functions

Finally, we must show that any function \(F\) can be expressed as the sum of an odd function and an even function. Consider expressing \(F(x)\) as:\[F(x) = E(x) + O(x)\]where \(E(x)\) is even and \(O(x)\) is odd. Set:\[E(x) = \frac{F(x) + F(-x)}{2}\]\[O(x) = \frac{F(x) - F(-x)}{2}\]Clearly, \(E(x)\) is even and \(O(x)\) is odd as proven above. Adding them, we have:\[E(x) + O(x) = \frac{F(x) + F(-x)}{2} + \frac{F(x) - F(-x)}{2} = \frac{2F(x)}{2} = F(x)\] Thus, \(F\) can be expressed as the sum of an odd and an even function.

Key Concepts

Odd Function

An odd function is a fascinating type of function in mathematics with a particular symmetry property. Specifically, for a function \(f(x)\) to be considered odd, it must satisfy the condition \(f(-x) = -f(x)\) for all \(x\) in its domain. This symmetry implies that the graph of an odd function is symmetric about the origin. If you reflect the graph over both the x-axis and the y-axis, it will look unchanged.

To prove that \(F(x) - F(-x)\) is an odd function, we start by considering \(f(x) = F(x) - F(-x)\). When we plug \(-x\) into this function, we have:

• \(f(-x) = F(-x) - F(x)\)

This is equivalent to \(-f(x)\), which confirms that \(f(x)\) is indeed an odd function, as it's equal to its own negative when \(-x\) is plugged in.

Even Function

Even functions are another significant category of functions characterized by their own unique symmetry. For a function \(g(x)\) to be classified as even, it must meet the criterion \(g(-x) = g(x)\) for all \(x\) in its domain. The symmetry of even functions implies that their graph remains unchanged if reflected across the y-axis.

Let's demonstrate that \(F(x) + F(-x)\) is an even function. Define \(g(x) = F(x) + F(-x)\). When computing \(g(-x)\), we find:

• \(g(-x) = F(-x) + F(x)\)

By recognizing that addition is commutative, this simplifies to \(g(-x) = g(x)\), thereby proving that \(g(x)\) is an even function.

Function Properties

Understanding the properties of functions is crucial for analyzing and manipulating them. Functions can have a variety of properties, but symmetry, leading to odd or even characterizations, stands out for its useful insights into graphs and transformations.

When working with a given function \(F\), it's often helpful to decompose it into simpler pieces, such as the sum of an odd part and an even part. To accomplish this, we use:

• Even part: \(E(x) = \frac{F(x) + F(-x)}{2}\)
• Odd part: \(O(x) = \frac{F(x) - F(-x)}{2}\)

These components share the domain of \(F\) and allow us to express \(F(x) = E(x) + O(x)\), enabling further analysis and applications, especially in contexts like Fourier series.

Mathematical Proof

A mathematical proof is a logical argument that establishes the truth of a mathematical statement beyond any doubt. It involves a series of deductive reasoning steps that build from agreed-upon axioms and previously established theorems.

Proving properties, such as whether a function is odd or even, not only consolidates our understanding but also deepens our grasp of the function's behavior. With \(F(x) - F(-x)\) and \(F(x) + F(-x)\), proofs rely on checking definitions:

• An odd function: confirm using \(f(-x) = -f(x)\)
• An even function: ensure \(g(-x) = g(x)\)

Following each step carefully with logical reasoning ensures the validity and builds confidence in the result.