Question

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

Short answer

The product of an odd function and an even function is odd.

Step-by-step solution

Definitions of Even and Odd Functions

Even function, \(f(x)\), fulfills the condition \(f(x) = f(-x)\) and the Odd function, \(g(x)\), satisfies the property \(g(x) = -g(-x)\), for all \(x\) in the domain of the function.

Form The Product of The Functions

Let's define \(h(x)\) as the product of the even function \(f(x)\) and the odd function \(g(x)\). So the function \(h(x) = f(x) \cdot g(x)\)

Substitute the Properties of Even and Odd Functions in \(h(x)\)

\(h(-x) = f(-x) \cdot g(-x)\). As \(f(-x) = f(x)\) and \(g(-x) = -g(x)\), then \(h(-x) = f(x) \cdot -g(x) = -f(x) \cdot g(x) = -h(x)\)

Check the Requirement for An Odd Function

The function \(h(x)\) satisfies the condition \(h(-x) = -h(x)\), which means it is odd.