Which of the following functios is odd and which one is even?

Question

(a) #y=1/(x^2+4)#
(b) #y=x/(x^3+3)#
(c) #y=2^x+2^(-x)#

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Answer 1 · 2017-10-18T07:26:22

Please see below.

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

if #f(x)# is odd, we have #f(-x)=-f(x)#

and if neither #f(-x)!=-f(x)# nor #f(-x)!=f(x)#, it is neither odd nor even.

(a) As #y=1/(x^2+4)#, #f(-x)=1/((-x)^2+4)=1/(x^2+4)=f(x)#

hence #y=1/(x^2+4)# is even.

(b) As #y=x/(x^3+3)#, #f(-x)=(-x)/((-x)^3+3)=-x/(-x^3+3)=x/(x^3-3)#,

which is neither #f(x)# nor #-f(x)#.

Hence #y=x/(x^3+3)# is neither odd nor even.

(c) As #y=2^x+2^(-x)#, #f(-x)=2^(-x)+2^(-(-x))=2^(-x=f(x)#

hence #y=2^x+2^(-x)# is even.