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