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"):}`

Which of the following functions has (have) y-symmetry or origin symme

1.	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"):}`
Answer» (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.