Consider to example odd functions and multiply them.
Is your friend correct? No Explanation: See solution.
Practice makes perfect
To test our friend's claim, we will calculate the product between two example odd functions. Recall that functions of the form y=x^n are odd in cases where the exponent is an odd number. Let's consider two odd functions.
f(x)=x^3andg(x)=x^5
Let h(x) be f(x)g(x). Let's find the equation for h(x).
Finally, let's calculate - h(x).
h(x)=x^8 ⇔ - h(x)= - x^8
Since h(- x)=x^8 and - h(x)=- x^8, we have that h(- x) ≠- h(x). Therefore, the product of the odd functions f and g is not an odd function. Our friend is not correct. The product of two odd functions is not always an odd function.
Extra
Can the product be an odd function?
In our solution we showed two odd functions for which the product is not odd.
Were we lucky to find this example, or is this always the case?
Let's check this using the definition of odd functions.
fis odd &⇔ f(- x)=- f(x)
gis odd &⇔ g(- x)=- g(x)
Let's use this definition to check the product at - x.
We can see that the result does not match the definition of an odd function. In fact, we obtained that the product of two odd functions is always an even function.
