Which of the following functions has (have) y-symmetry or origin symmetry?
(i) ` f(x) = x^(2) sin x" "(ii) f(x) = log (x+sqrt(1+x^(2)))`
(iii)` f(x) = (e^(x)+e^(-x))/2" "(iv) f(x) = {{:(0", ""If x is rational "),(1", " "If x is irrational"):}`

Explanation

(i) ` f(x) = (-x)^(2) sin (-x) =- x^(2) sin x =- f(x)`, hence function has origin symmetry
(ii) `f(-x) = log (-x+sqrt(1+(-x)^(2)))`
`= log(((-x+sqrt(1+x^(2)))(x+sqrt(1+x^(2))))/((x+sqrt(1+x^(2)))))`
`= log(1/(x+sqrt(1+x^(2))))`
` = - log(x+sqrt(1+x^(2)))`
=- f(x), hence function has origin symmetry.
(iii) `f(-x) = (e^(-x)+e^(-(-x)))/2 =(e^(-x)+e^(x))/2 = f(x)`,hence function has y-symmetry.
(iv) ` f(x) ={{:(0", ""If x is rational"),(1", ""If x is irrational"):}`
`rArr f(-x) ={{:(0", ""If -x is rational"),(1", ""If -x is irrational"):}`
`={{:(0", ""If x is rational"),(1", ""If x is irrational"):}`
= f(x)
Hence functions has y-symmetry.