Question

Which of the following function is (are) even, odd, or neither? `f(x)=x^2sinx` `f(x)=sqrt(1+x+x^2)-sqrt(1-x+x^2)` `f(x)=log((1-x)/(1+x))` `f(x)=log(x+sqrt(1+x^2))` `f(x)=sinx-cosx` `f(x)=(e^x+e^(-x))/2`

Answer 1

(i) `f(-x)=(-x)^(2) sin(-x)= -x^(2)sinx= -f(x)`
Hence, f(x) is odd.
(ii) `f(-x)=sqrt(1+(-x)+(-x)^(2))-sqrt(1-(-x)+(-x)^(2)) `
`=sqrt(1-x+x^(2))-sqrt(1+x+x^(2))`
`= -f(x)`
Hence, f(x) is odd.
(iii) `f(-x)=log{(1-(-x))/(1+(-x))}`
` log((1+x)/(1-x))`
`= -f(x)`
Hence, f(x) is odd.
(iv) `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))))= -f(x)`
Hence, f(x) is odd.
(v) `f(-x)=sin(-x)-cos(-x)= -sinx-cosx`
Clearly, `f(-x) ne f(x) " and " f(-x) ne -f(x).`
Hence, f(x) is neither even nor odd.
(vi) `f(-x)=(e^(-x)+e^(-(-x)))/(2)=(e^(-x)+e^(x))/(2)=f(x)`
Hence, f(x) is even.