Question

Let $f ( x )=\sin\, x$ and $g ( x )=\log _{ e }| x | .$ If the ranges of the composition functions fog and gof are $R _{1}$ and $R _{2}$, respectively, then

• A. $R _{1}=\{ u :-1 \leq u <1\}, R _{2}=\{ v :-\infty < v < 0\}$
• B. $R_{1}=\{u:-\infty < u < 0\}, R_{2}=\{v:-\infty < v < 0\}$
• C. $R_{1}=\{u:-1 < u < 1\}, R_{2}=\{v:-\infty < v < 0\}$
• D. $R_{1}=\{u:-1 \leq u \leq 1\}, R_{2}=\{v:-\infty < v \leq 0\}$

Answer 1

Answer: (D)

We have fog $(x)=f(g(x))=\sin \left(\log _{e}|x|\right)$.
$\log _{ e }| x |$ has range $R$, for which $\sin \left(\log _{ e }| x |\right) \in[-1, 1]$
Therefore, $R _{1}=\{ u :-1 \leq u \leq 1\}$
Also, gof $(x)=g(f(x))=\log _{e}|\sin\, x|$
$\because 0 \leq|\sin \,x | \leq 1$ or $-\infty < \log _{ e }|\sin \,x | \leq 0$
or $R _{2}=\{ v :-\infty< v \leq 0\}$