Find the domain of each of the following functions given by
(a) `f(x)=(1)/(sqrt(1-cosx))`
(b) `f(x)=(1)/(sqrt(x+|x|))`
(c) `f(x)=x|x|`
(d) `f(x)=(x^(3)-x+3)/(x^(2)-1)`
`(e) f(x)=(3x)/(28-x)`

Explanation

(i) We have `f(x)=(1)/(sqrt(1-cosx))`
`:. 1-le cosx le1`
`rArr -1le-cosxle1`
`rArr 0le1-cosxle2`
So, f(x) is defined, if `1-cosx ne 0`
`cosx ne 1`
`x ne 2npiAAninZ`
`:.` Domain of `f=R-{2npi : n in Z}`
(ii) We have, `f(x)=(1)/(sqrt(x+|x|))`
`:. x+|x|=x-x=0,x lt0`
`=x+x=2x,xge0`
Hence, f(x) is defined , if `xgt0`
` :.` Domain of `f=R^(+)`
(iii) We have, `f(x)=x|(x|)`
clearly, `f(x)` is defined for any `x in R`.
(iv) We have, `f(x)=(x^(3)-x+3)/(x^(2)-1)`
f(x) is not defined, If `x^(2)-1=0`
`rArr (x-1)(x+1)=0`
`rArr x=-1,1`
`:.` Domain of `f=R-{-1,1}`
(v) We have, `f(x)=(3x)/(28-x)`
Clearly, f(x) is defined, if `28-x ne0`
`rArr x ne28`
`:.` Domain of `f=R-{28}`