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QUESTION

# When you multiply two odd or two even functions, what type of function will you get?

Always even (unless its domain is empty).

color(white)()Odd times odd

Suppose f(x) and g(x) are odd functions and h(x) = f(x)g(x)

By definition:

f(-x) = -f(x) and g(-x) = -g(x) for all x

So:

h(-x) = f(-x)g(-x) = (-f(x))(-g(x)) = f(x)g(x)

= h(x) for all x

So h(x) is even.

color(white)()Even times even

Now suppose that f(x) and g(x) are even functions and h(x) = f(x)g(x)

By definition:

f(-x) = f(x) and g(-x) = g(x) for all x

So:

h(-x) = f(-x)g(-x) = f(x)g(x) = h(x) for all x

So h(x) is even.

color(white)()Exception

If the domains of f(x) and g(x) do not intersect, then their product f(x)g(x) has an empty domain, so is the empty function. The empty function probably does not count as odd or even.

For example:

Let f(x) = cos^(-1)(x) and g(x) = sec^(-1)(x/2).

Then the domain of f(x) is [-1, 1] and the domain of g(x) is (-oo, -2] uu [2, oo). They are both even functions.

The domain of f(x)g(x) is [-1, 1] nn ((-oo, 2] uu [2, oo)) = O/