Problem 48 - Calculus with Concepts in Calculus

Question

Let \(f\) and \(g\) be even functions. a. Show that \(f+g\) is an even function. b. Show that \(f g\) is an even function.

Step-by-Step Solution

Answer

Both \(f+g\) and \(fg\) are even functions as they satisfy the symmetry condition of even functions.

Step 1: Understand the Definition of Even Function

A function is called even if for all values in its domain, the property \(f(x) = f(-x)\) holds. This means that the function is symmetric with respect to the y-axis.

Step 2: Prove f + g is Even

For functions \(f\) and \(g\) that are even, we know that \(f(x) = f(-x)\) and \(g(x) = g(-x)\). To show \(f+g\) is even, check \((f+g)(x) = (f+g)(-x)\):\[(f+g)(x) = f(x) + g(x)\]\[(f+g)(-x) = f(-x) + g(-x) = f(x) + g(x)\]Since \((f+g)(x) = (f+g)(-x)\), \(f+g\) is even.

Step 3: Prove f \( \cdot \) g is Even

For the product \(f \cdot g\), check if \((f\cdot g)(x) = (f\cdot g)(-x)\):\[(f \cdot g)(x) = f(x) \cdot g(x)\]\[(f \cdot g)(-x) = f(-x) \cdot g(-x) = f(x) \cdot g(x)\]Since \((f\cdot g)(x) = (f\cdot g)(-x)\), \(f \cdot g\) is also even.

Key Concepts

Function Symmetry

Understanding function symmetry is key to solving many problems in mathematics, especially when dealing with even and odd functions. When we talk about the symmetry of a function, we often refer to how a function looks the same on either side of a specific axis. For even functions, this axis is the y-axis.
Given a function \( f \), it is called even if it satisfies the condition \( f(x) = f(-x) \) for every \( x \) in its domain. This means that if you were to fold the graph of the function along the y-axis, both halves would match perfectly.
Many common mathematical functions are even. For example, consider \( f(x) = x^2 \). Here, you can see that \( f(x) = f(-x) \) because \( x^2 = (-x)^2 \), showcasing the even nature of this particular function. Even functions reflect their values symmetrically around the y-axis, making them useful in many symmetrical applications in real-world scenarios.

Function Addition

Function addition involves combining two functions to form a new function. When we have two even functions, say \( f \) and \( g \), and we add them together, the result \( f + g \) is also an even function. This happens because both functions have the same symmetry property, \( f(x) = f(-x) \) and \( g(x) = g(-x) \).
To see why \( f+g \) is also even, consider the following:

• The function \( f+g \) at a point \( x \) is defined as \( (f+g)(x) = f(x) + g(x) \).
• At point \( -x \), \( (f+g)(-x) = f(-x) + g(-x) \).
• Since we know both \( f(x) = f(-x) \) and \( g(x) = g(-x) \), it implies that \( f(-x) + g(-x) = f(x) + g(x) \).

This confirms that \( (f+g)(x) = (f+g)(-x) \), making \( f+g \) an even function. This property of symmetry retains itself even when functions are summed, provided both are even.

Function Multiplication

Function multiplication is an operation where we multiply two functions together. When both functions are even, like \( f \) and \( g \) from our exercise, their product \( f \cdot g \) is also even. The symmetry of the functions is preserved through multiplication.
To demonstrate this:

• Consider the product at \( x \): \( (f \cdot g)(x) = f(x) \cdot g(x) \).
• Similarly, consider the product at \( -x \): \( (f \cdot g)(-x) = f(-x) \cdot g(-x) \).
• Since \( f(x) = f(-x) \) and \( g(x) = g(-x) \), it follows that \( f(-x) \cdot g(-x) = f(x) \cdot g(x) \).

Thus, \( (f \cdot g)(x) = (f \cdot g)(-x) \) confirms that \( f \cdot g \) is an even function. Multiplying two even functions results in the preservation of their symmetry properties, making multiplication a reliable operation for combining these types of functions.