Answer
Prove that the product of two even functions is even :
1. $h(x)=f(x)g(x)$
2. $h(-x)=f(-x)g(-x)$
3. $h(-x)=f(x)g(x)$
4. $h(-x)=h(x)$
1. Let function $h$ be the product of even functions $f$ and $g$.
2. Evaluate for $h(-x)$
3. Since $f$ and $g$ are even, $f(-x)=f(x)$ and $g(-x)=g(x)$
4. Transitive property of equality
Therefore the product of two even functions is even.
Prove that the product of two odd functions is even:
1. $h(x)=f(x)g(x)$
2. $h(-x)=f(-x)g(-x)$
3. $h(-x)=-f(x)*-g(x)$
4. $h(-x)=f(x)g(x)$
5. $h(-x)=h(x)$
1. Let function $h$ be the product of odd functions $f$ and $g$.
2. Evaluate for $h(-x)$
3. Since $f$ and $g$ are odd, $f(-x)=-f(x)$ and $g(-x)=-g(x)$
4. Simplify
5. Transitive property of equality
Therefore the product of two odd functions is an even function.
Work Step by Step
See the proofs above
