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Chapter 0: Problem 102

Proof 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 odd.

Step by step solution

01

Define the functions

Let's denote an odd function as \( O(x) \) and an even function as \( E(x) \). The product of the two will be \( O(x)E(x) \).
02

Identify the key properties

The key characteristics of an odd function \( O(x) \) is that \( O(-x) = -O(x) \) and for an even function \( E(x) \) is that \( E(-x) = E(x) \).
03

Substitute the opposite of x

Now, let's substitute -x into the product. We get \( O(-x)E(-x) \) replacing \( -x \) in each function with their respective properties, we get \( -O(x)E(x) \).
04

Assert and generalize

As \( -O(x)E(x) \) equates to the negation of the original product \( O(x)E(x) \), we can confirm that the product of the odd function and the even function is indeed an odd function.

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