Let Ex be a function with even symmetry and Ox be a function

Let E(x) be a function with even symmetry and O(x) be a function with odd symmetry. Use the definitions of even and odd symmetry to prove that F(x)=(E(x)+O(x))(E(x)-O(x)) has even, odd, or no symmetry. You must show that F(x) is even,odd, or neither for EVERY VALUE OF X YOU USE.

Solution

A function with even symmetry will show no change in it\'s value with a change in the sign of the variable, whereas in an odd function a change in the sign of the variable will be reflected in the function\'s value.

E (x) is a function with even symmetry so, E (-x)= E (x)and

O (x) is a function of odd symmetry, hence O (-x)=-O (x)

F (x)= [E (x)+O (x)][E (x)-O (x)]

F (-x)= [E (-x)+O (-x)][E (-x)-O (-x)]

=[E (x)+ (-O (x))][E (x)-( -O (x))]

= [E (x)-O (x)][E (x)+O (x)]

= [E (x)+O (x)][E(x)-O (x)]

= F (x)

Since, F (-x)= F (x) , for negative x, thus F (x) is a function with even symmetry.