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

Nov 14, 2015

odd

#### Explanation:

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

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

Let $h \left(x\right) = f \left(x\right) g \left(x\right)$

Then:

$h \left(- x\right) = f \left(- x\right) g \left(- x\right) = \left(- f \left(x\right)\right) g \left(x\right) = - \left(f \left(x\right) g \left(x\right)\right) = - h \left(x\right)$

for all $x$

That is $h \left(x\right)$ is odd.