Which of the following functions has (have) y-symmetry or origin symmetry?
(i) ` f(x) = x^(2) sin x" "(ii) f(x) = log ((1-x)/(1+x))`
(iii)` f(x) = x/(e^(x)-1)+x/2 + 1`

Explanation

(i) `f(-x) = (-x)^(2) sin (-x) =- x^(2) sin x =- f(x)`, hence the function has origin symmetry
(ii) `f(-x)= log((1-(-x))/(1+(-x))) = log ((1+x)/(1-x))=- f(x)`hence the function has origin symmetry
(iii) `f(x) = x/(e^(x)-1)+x/2 + 1`
` rArr f(-x) = (-x)/(e^(-x)-1)-x/2 + 1`
` = (xe^(x))/(e^(x)-1) - x/2 + 1`
`=(xe^(x)-x+x)/(e^(x)-1) - x/2+1`
` = x+x/(e^(x)-1)-x/2 + 1`
` = x/(e^(x)-1) + x/2 + 1`
= f(x)
Hence the function has y-symmetry.