Problem 22 - Calculus Early Transcendentals: Pearson New International Edition

Question

State whether each of the following is an odd function, an even function, or neither. Prove your statements. (a) The sum of two even functions (b) The sum of two odd functions (c) The product of two even functions (d) The product of two odd functions (e) The product of an even function and an odd function

Step-by-Step Solution

Answer

(a) Even, (b) Odd, (c) Even, (d) Even, (e) Odd.

Step 1: Define Even and Odd Functions

An **even function** satisfies the property \( f(-x) = f(x) \) for all \( x \). An **odd function** satisfies \( g(-x) = -g(x) \) for all \( x \). If neither condition is satisfied, then the function is neither even nor odd.

Step 2: Analyze (a) Sum of Two Even Functions

Let the even functions be \( f(x) \) and \( g(x) \). Their sum is \( h(x) = f(x) + g(x) \). To determine if \( h(x) \) is even, check \( h(-x) = f(-x) + g(-x) = f(x) + g(x) = h(x) \). Thus, a sum of two even functions is an even function.

Step 3: Analyze (b) Sum of Two Odd Functions

Let the odd functions be \( f(x) \) and \( g(x) \). Their sum is \( h(x) = f(x) + g(x) \). Check \( h(-x) = f(-x) + g(-x) = -f(x) - g(x) = -(f(x) + g(x)) = -h(x) \). Thus, a sum of two odd functions is an odd function.

Step 4: Analyze (c) Product of Two Even Functions

Let the even functions be \( f(x) \) and \( g(x) \). Their product is \( h(x) = f(x)g(x) \). Check \( h(-x) = f(-x)g(-x) = f(x)g(x) = h(x) \). Therefore, the product of two even functions is an even function.

Step 5: Analyze (d) Product of Two Odd Functions

Let the odd functions be \( f(x) \) and \( g(x) \). Their product is \( h(x) = f(x)g(x) \). Check \( h(-x) = f(-x)g(-x) = (-f(x))(-g(x)) = f(x)g(x) = h(x) \). Thus, the product of two odd functions is an even function.

Step 6: Analyze (e) Product of an Even and an Odd Function

Let the even function be \( f(x) \) and the odd function be \( g(x) \). Their product is \( h(x) = f(x)g(x) \). Check \( h(-x) = f(-x)g(-x) = f(x)(-g(x)) = -f(x)g(x) = -h(x) \). So, the product of an even and odd function is an odd function.

Key Concepts

Even Functions

Understanding the nature of even functions can greatly help in recognizing patterns in calculus. An even function is characterized by the property that its graph is symmetric with respect to the y-axis. This means that for any function \( f(x) \), if you plug in \(-x\), the function's value remains unchanged, i.e., \( f(-x) = f(x) \) for all \( x \). This symmetry can simplify calculations, especially when dealing with integrals over symmetric intervals.

Examples of even functions include:

• \( f(x) = x^2 \)
• \( g(x) = \cos(x) \)

In practical applications, recognizing even functions allows for efficient computational methods in problem-solving due to their inherent symmetry.

Odd Functions

Odd functions are fascinating due to their unique symmetry about the origin. This means that an odd function, \( g(x) \), satisfies the condition \( g(-x) = -g(x) \) for all \( x \). The graph of an odd function reflects across the origin, exhibiting point symmetry. This property is particularly useful in calculus when dealing with definite integrals because the integral of an odd function over symmetric limits is always zero.

Some examples of odd functions are:

• \( g(x) = x^3 \)
• \( h(x) = \sin(x) \)

This insight into odd functions can simplify the analysis of mathematical and real-world problems by leveraging their symmetry.

Function Properties

Exploring the properties of functions lays the groundwork for deeper understanding in calculus. When analyzing functions, it's crucial to recognize whether they possess either even or odd properties because this informs their behavior and can simplify further calculations.

Key properties include:

• Symmetry: Even functions are symmetric about the y-axis, while odd functions are symmetric about the origin.
• Integration: For symmetric limits, the integral of an even function relies solely on the positive half of the function, and odd functions contribute zero to the integral.

Understanding these properties can transform how we approach complex mathematical problems, applying specific techniques suited for even or odd functions.

Sum of Functions

Combining functions through addition can result in new functions that inherit properties from their components. The sum of two functions is a straightforward operation, but its resulting properties can vary depending on the initial functions involved.

Consider:

• The sum of two even functions remains even. This occurs because each individual function's symmetry is preserved in their combination.
• The sum of two odd functions results in another odd function, maintaining the origin symmetry in their sum.

These rules make it easier to predict the properties of more complex function combinations, streamlining the process of evaluation and analysis.

Product of Functions

Function multiplication introduces new possibilities and complexities due to the interaction between individual function characteristics. When you multiply functions, the resulting function's properties depend significantly on the nature of the original functions.

Key points include:

• The product of two even functions is itself even, as both functions contribute to retaining y-axis symmetry.
• Two odd functions multiplied together yield an even function because the negative signs cancel out, restoring symmetry.
• An even function multiplied by an odd function results in an odd function. The lack of symmetry in one and presence in the other leads to the characteristic asymmetry of the odd function.

These insights are crucial in systematically understanding how operations on functions influence their overall behavior, which is pivotal in problem-solving.