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Chapter 0: Problem 92

Prove that the product of an odd function and an even function is odd.

Short Answer

Expert verified
The product of an odd function and an even function is an odd function because it follows the negative symmetry about the origin that is characteristic of all odd functions.

Step by step solution

01

Define the Even Function

An even function, h(x), is one for which h(-x) = h(x) for all x in the function's domain.
02

Define the Odd Function

An odd function, g(x), is one for which g(-x) = -g(x) for all x in the function's domain.
03

Express the Product Function

Let p(x) = g(x) * h(x) be the product of the odd and even function respectively.
04

Substitute the Opposites

We need to prove that p(-x) = -p(x). Substituting -x into p(x), we get p(-x) = g(-x) * h(-x).
05

Apply Even and Odd Function Properties

Substitute both the even and odd properties into this equation. Thus, p(-x) = -g(x) * h(x); because of the even function property, h(-x) becomes h(x).
06

Comparison

This can be written as -1* g(x) * h(x) which is -p(x).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Product of Functions
When dealing with functions, one important operation is taking the product of two functions. This means creating a new function that is obtained by multiplying the outputs of two existing functions for each input value.
For example, if you have two functions \( g(x) \) and \( h(x) \), the product of these functions, denoted as \( p(x) \), is defined by the equation \( p(x) = g(x) \cdot h(x) \).
This operation results in a combined function that exhibits behaviors influenced by both original functions. Understanding the characteristics of the product function helps in predicting its behavior under various conditions.
Function Properties
Functions can have special properties, such as being odd or even. These properties significantly affect the behavior and transformation of functions. Understanding these will aid in performing operations like function multiplication.
  • Even function: A function \( h(x) \) is even if it satisfies \( h(-x) = h(x) \) for all \( x \) in its domain. This symmetry indicates that the function remains unchanged when reflected across the y-axis.
  • Odd function: A function \( g(x) \) is odd if it satisfies \( g(-x) = -g(x) \) for all \( x \). This property indicates a rotational symmetry about the origin, where reversing the sign of the input reverses the sign of the output.
Recognizing these properties helps in reasoning about the behavior of function combinations, like the product of an odd and an even function.
Proving Mathematical Statements
To prove mathematical statements, especially those involving properties of functions, careful logical steps are essential. Let's look at the proof for the product of an odd and an even function being odd.
Given an odd function \( g(x) \) and an even function \( h(x) \), you define their product as \( p(x) = g(x) \times h(x) \).
To demonstrate that \( p(x) \) is odd, you need to show \( p(-x) = -p(x) \).
  • Start by substituting \(-x\) into the product: \( p(-x) = g(-x) \times h(-x) \).
  • Apply the properties of the functions: the even function ensures \( h(-x) = h(x) \), and the odd function gives \( g(-x) = -g(x) \).
  • This results in \( p(-x) = (-g(x)) \times h(x) = - (g(x) \times h(x)) = -p(x) \), confirming \( p(x) \) is odd.
Such proofs reveal the logical beauty in mathematics, showcasing how defined properties lead to new insights.

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