If `f(x)=sin(log)_e{(sqrt(4-x^2))/(1-x)}` , then the domain of `f(x)` is _____ and its range is __________.

Explanation

We have `f(x)=sin(log_(e)((sqrt(4-x^(2)))/(1-x)))`
We must have `4-x^(2) gt0 " and " 1-x-0`
`implies x^(2) lt 4 " and " x lt 1`
`implies -2 lt x lt 2 " and " x lt 1`
`implies -2 lt x lt 1`
Thus, domain of f(x) is `(-2,1)`
When x approaches to 1 from its left-hand side `(sqrt(4-x^(2)))/(1-x)`
approaches to infinity.
When x approaches to -2 from its right-hand side `(sqrt(4-x^(2)))/(1-x)`
approaches to zero.
Also, `(sqrt(4-x^(2)))/(1-x)` exists continuously for ` x in (-2,1)`.
Thus `0 lt (sqrt(4-x^(2)))/(1-x) lt oo`
`implies -oo lt "log"_(e) (sqrt(4-x^(2)))/(1-x) lt oo`
`implies sin("log"_(e)(sqrt(4-x^(2)))/(1-x)) in [-1 ,1]`