Is the product of an odd function and an even function odd or even?

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Answer 1 · 2015-11-14T11:12:27

odd

Suppose $f (x)$ $f(x)$ is odd and $g (x)$ $g(x)$ is even.

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

Let $h (x) = f (x) g (x)$ $h(x) = f(x)g(x)$

Then:

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

for all $x$ $x$

That is $h (x)$ $h(x)$ is odd.