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Range of a Function

Let fx=sin x,...

Question

Let $f (x) = sin x, g (x) = {log}_{e} | x |$ . If the ranges of the composite functions $f o g$ and $g o f$ are $R_{1}$ and $R_{2}$ respectively, then

R_{1} = {μ : - 1 \leq μ < 1}

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R_{2} = {ϑ : - \infty < ϑ < 0}

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R_{1} = {μ : - 1 \leq μ \leq 1}

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R_{2} = {ϑ : - \infty < ϑ \leq 0}

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Solution

The correct option is D $R_{2} = {ϑ : - \infty < ϑ \leq 0}$
$\begin{matrix} fog (x) = f ({log}_{e} | x |) = sin ({log}_{e} | x |) ∴ R_{1} = {μ : - 1 \leq μ \leq 1} gof (x) = g (sin x) = {log}_{e} | sin x | Now | sin x | \in [0, 1] \Rightarrow - \infty < log | sin x | \leq 0 \end{matrix}$

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