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