Problem 49 - Calculus with Concepts in Calculus

Question

Let \(f\) be even and \(g\) odd. Show that \(f g\) is an odd function.

Step-by-Step Solution

Answer

The function \( fg \) is odd because \( (fg)(-x) = -(fg)(x) \).

Step 1: Define Even and Odd Functions

A function \( f(x) \) is even if \( f(-x) = f(x) \) for all \( x \). A function \( g(x) \) is odd if \( g(-x) = -g(x) \) for all \( x \).

Step 2: Determine Expression for \( f g \) at \( -x \)

We want to determine \( (fg)(-x) \). Start by substituting into the expression: \( (fg)(-x) = f(-x)g(-x) \).

Step 3: Utilize Properties of Even and Odd Functions

Since \( f \) is even, \( f(-x) = f(x) \). Since \( g \) is odd, \( g(-x) = -g(x) \). Substitute these into the expression from Step 2: \( (fg)(-x) = f(x) (-g(x)) = -f(x)g(x) \).

Step 4: Show \( f g \) is Odd

Compare \( (fg)(-x) \) with \( -(fg)(x) \), which is \( -f(x)g(x) \). Since both are equal, we have \( (fg)(-x) = -(fg)(x) \). Therefore, by definition, \( fg \) is an odd function.

Key Concepts

Even Function

An even function is one of those ways nature likes to keep things symmetric and balanced. Think of an even function like seeing your reflection perfectly mirrored in a serene lake. Mathematically, a function \( f(x) \) is called even if its values remain unchanged when you flip the input signs. In other words, \( f(x) = f(-x) \) for any value of \( x \). This property ensures that the function is symmetrical with respect to the y-axis.

Here are some key points about even functions:

• They exhibit a symmetric graph about the y-axis.
• Common examples include \( y = x^2 \) and \( y = \cos(x) \).
• Their behavior at positive and negative inputs are identical.

Understanding the symmetry helps students recognize even functions easily by just observing their graphs or equations.

Odd Function

Odd functions are a bit of a twisty concept, full of turns and flips. When you think of an odd function, imagine something that somersaults across the origin. In formal terms, a function \( g(x) \) is considered odd if substituting \( -x \) into the function results in the negative of the original function, that is \( g(-x) = -g(x) \). This characteristic gives the function a rotational symmetry about the origin.

Highlighting odd function properties:

• Graphically, these functions have origin symmetry which means rotating them 180 degrees will map the function onto itself.
• Examples include \( y = x^3 \) and \( y = \sin(x) \).
• Their symmetry is opposite to that of even functions, offering a "negative" reflection for corresponding x-values.

Recognizing an odd function helps understand how such functions behave when subjected to various transformations and applications.

Properties of Functions

Both even and odd functions present inclusive features that make function analysis straightforward when dealing with complex operations like multiplication or addition of functions. The properties of these functions stem from their definitions and can greatly determine the symmetry and behavior of the resulting functions.

Key properties include:

• Multiplying an even function with another even function results in an even function because both retain their symmetry around the y-axis.
• Multiplying an odd function with another odd function gives an even function due to their rotational symmetries effectively canceling out.
• Multiplying an even function with an odd function results in an odd function, which showcases the influence of each function's properties.
• Even though the properties of being even or odd seem strict, they can play in advantage to predict the result of function combinations.

These properties help simplify how to handle functions in many mathematical contexts, establishing a clear path in analyzing more intricate function scenarios.