# If f(x) is an even function and g(x) is an odd function, how do you prove that h(x) = f(x) * g(x) is odd?

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To prove that ( h(x) = f(x) \cdot g(x) ) is odd when ( f(x) ) is an even function and ( g(x) ) is an odd function, you need to show that ( h(-x) = -h(x) ) for all ( x ) in the domain.

Given that ( f(x) ) is an even function, ( f(-x) = f(x) ).

And since ( g(x) ) is an odd function, ( g(-x) = -g(x) ).

Now, evaluate ( h(-x) = f(-x) \cdot g(-x) ).

Substitute the expressions for ( f(-x) ) and ( g(-x) ) with their respective properties.

[ h(-x) = f(x) \cdot (-g(x)) = - (f(x) \cdot g(x)) = -h(x) ]

Thus, ( h(x) = f(x) \cdot g(x) ) is an odd function.

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