Let f(x) = sin x and g(x) = ln|x|. If the ranges of the composite functions fog and gof are $R_{1}$ and $R_{2}$ respectively, then

R_{1} = {u : - 1 < u < 1}, R_{2} = {v : - \infty < v < 0}

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R_{1} = {u : - \infty < u \leq 0}, R_{2} {v : 1 \leq v \leq 1}

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R_{1} = {u : - 1 < u < 1}, R_{2} = {v : - \infty > v < 0}

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R_{1} = {u : - 1 \leq u \leq 1}, R_{2} = {v : - \infty < v \leq 0}

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Solution

The correct option is D $R_{1} = {u : - 1 \leq u \leq 1}, R_{2} = {v : - \infty < v \leq 0}$
$f o g = f (g (x)) = s i n l n | x | a n d g o f = g (f (x)) = l n | s i n x |$
$∴ R_{1} = [- 1, 1] a n d R_{2} = (- \infty, 0] .) (∵ | s i n x | ϵ [0, 1])$

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