Question

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

Answer 1

Always even (unless its domain is empty).

Explanation:

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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.

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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.

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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/`