If #f(x)# is even and #g(x)# is even, then find whether following functions are odd or even?

Question

(a) #h(x)=f(x)g(x)#
(b) #h(x)=f(x)/g(x)#
(a) #h(x)=f(x)(g(x))^2#

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Answer 1 · 2017-10-18T07:05:24

Please see below.

As #f(x)# is even, we have #f(-x)=f(x)#

and as #g(x)# is odd, we have #g(-x)=-g(x)#

Therefore,

(a) if #h(x)=f(x)g(x)#,

then #h(-x)=f(-x)* g(-x)=f(x)*(-g(x))#

= #-f(x)g(x)=-h(x)#,

hence #h(x)# is odd.

(b) if #h(x)=f(x)/g(x)#,

#h(-x)=f(-x)/g(-x)=f(x)/(-g(x))=-f(x)/g(x)=-h(x)#

hence #h(x)# is odd.

(c) if #h(x)=f(x)(g(x))^2#,

#h(-x)=f(-x)(g(-x))^2=f(x)(-g(x))^2#

= #f(x)(g(x))^2=h(x)#

hence #h(x)# is even.