Question

Even-Odd Show that the product of an even function and an odd function is an odd function.

Short answer

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

Step-by-step solution

Write down the definitions of even and odd functions

An even function is defined as a function that satisfies \(f(-x) = f(x)\) for all values of \(x\) in its domain. An odd function is a function that satisfies \(f(-x) = -f(x)\) for all values of \(x\) in its domain.

Define our even function \(f(x)\) and odd function \(g(x)\)

Let's consider \(f(x)\) as an even function and \(g(x)\) as an odd function. Meaning \(f(-x)=f(x)\) and \(g(-x)=-g(x)\). We are going to multiply these two functions.

Multiply even and odd functions

We have \(h(x) = f(x)g(x)\), where \(h(x)\) is the product of our even function \(f(x)\) and our odd function \(g(x)\).

Substitute \(x\) with \(-x\) in \(h(x)\)

If we substitute \(x\) with \(-x\) in \(h(x)\), we have \(f(-x)g(-x) = f(x)*(-g(x)) = -f(x)g(x) = -h(x)\). This relation, \(h(-x) = -h(x)\), shows that the function \(h(x)\) is odd.