Question

Products of Even and Odd Functions Answer the same questions as in exercise 102, except this time consider the product off and ginstead of sum.

Short answer

The product of two even function is even.

The product of two odd functions is even.

The product of one even and odd function is odd.

Step-by-step solution

Step 1. Answer the first question.

Let f and g be two even function and there product is h.

$$\begin{gathered} h\left(x\right)=f\left(x\right)\times g\left(x\right) \\ h(-x)=f(-x)\times g(-x) \\ =f\left(x\right)\times g\left(x\right) \\ =h\left(x\right) \end{gathered}$$

Hence, the product of two even function is even.

Step 2. Answer the second question.

Let f and g be two odd function and there product is h.

$$\begin{gathered} h\left(x\right)=f\left(x\right)\times g\left(x\right) \\ h(-x)=f(-x)\times g(-x) \\ =-f\left(x\right)\times -g\left(x\right) \\ =f\left(x\right)\times g\left(x\right) \\ =h\left(x\right) \end{gathered}$$

Hence, the product of two odd function is even.

Step 3. Answer the third question.

Let $f\left(x\right)$be even function and $g\left(x\right)$be odd function and be their product

$$\begin{gathered} h\left(x\right)=f\left(x\right)\times g\left(x\right) \\ h(-x)=f(-x)\times g(-x) \\ =f\left(x\right)\times -g\left(x\right) \\ =-\left[f\left(x\right)\times g\left(x\right)\right] \\ =h\left(x\right). \end{gathered}$$

So, the product of two odd function is odd.