Find the domain and range of following functions
(i) `f(x)=log_(e)(sinx)`
(ii) `f(x)=log_(3)(5-4x-x^(2))`

Explanation

(i) `f(x)=log_(e) sin x` is defined if `sin x in (0,1]`
` :. x in … (-4pi,-3pi)cup(-2pi,-pi) cup (0,pi) cup (2pi,3pi) cup (4pi,5pi)…`
Also, `0 lt sinx le 1`
` :. -oo lt log_(e)(sinx)le 0`
Thus, range is `(- oo,0]`
(ii) `f(x)=log_(3)(5-4x-x^(2))`
`=log_(3)(9-(x+2)^(2))`
f(x) is defined if `5-4x-x^(2) gt 0`
or `x^(2)+4x-5 lt 0`
`implies (x-1)(x+5) lt 0`
`implies -5 lt x lt 1`
So, domain is `(-5,1)`.
Also `9-(x+2)^(2) le 9`
Thus, `0 lt 9-(x+2)^(2) le 9`
`implies -oo lt log_(3)(9-(x+2)^(2)) le log_(3)9`
Hence, range of f(x) is `(-oo,2]`