If #u# and #v# are odd functions of #x#, what can be said about #(u+v)#,#(u-v)#,#(u/v)# and #u*v#? how do you find if they are Even, Odd or neither?
1 Answer
A) Sum = odd
B) Difference = neither even or odd function
C) Quotient = even
D) Product = even
Explanation:
If both
# u(-x)=-u(x) #
# v(-x)=-v(x) #
A) Sum:
# (u+v)(-x) = u(-x) + v(-x) #
# \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ = -u(x) - v(x) #
# \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ = -(u(x) + v(x)) #
# \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ = -(u+v)(x) # , an odd function
B) Difference:
# (u-v)(-x) = u(-x) - v(-x) #
# \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ = -u(x) + v(x) #
# \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ != -(u+v)(x) # , neither even or odd function
C) Quotient:
# (u/v)(-x) = (u(-x))/(v(-x)) #
# \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ = (-u(x))/(-v(x)) #
# \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ = (u(x))/(v(x)) # , an even function
D) Product:
# (u*v)(-x) = u(-x) * v(-x) #
# \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ = (-u(x)) * (-v(x)) #
# \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ = u(x)*v(x) # , an even function
