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Chapter 1: Problem 82

Use examples to hypothesize whether the product of an odd function and an even function is even or odd. Then prove your hypothesis.

Short Answer

Expert verified
Based on the validation carried out, it can be conclusively stated that the product of an even function and an odd function is an odd function.

Step by step solution

01

Example(for hypothese)

Let's consider an even function \(f(x) = x^2\) and an odd function \(g(x) = x^3\). Their product is \(h(x) = f(x)g(x) = x^5\), which is an odd function. Thus, the hypothetical conclusion is that the product of an odd function and an even function is an odd function.
02

Hypothesis Validation

The hypothesis can be generally validated using the definitions of even and odd functions. To do so, let's assume that \(f\) is an even function and \(g\) is an odd function, and their product is \(h = fg\). Then for every real number \(x\), there are two cases to be evaluated: \(h(-x) = f(-x)g(-x)\) and \(h(x) = f(x)g(x)\). By substituting the definition of even and odd functions into the equations respectively, we are able to get - \(h(-x) = f(x)(-g(x)) = -h(x)\). This indeed validates our hypothesis, meaning the product of an odd function and an even function is always an odd function.

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