Functions Question 705
Question: Let $ f(x)=sinx $ and $ g(x)=log_{e}| x |. $ If the ranges of the composition functions fog and gof are $ R_1 $ and $ R_2 $ , respectively, then
Options:
A) $ R_1=\{u:-1\le u \lt 1\}, R_2=\{v:-\infty \lt v \lt 0\} $
B) $ R_1=\{u:-\infty \lt u \lt 0\},R_2=\{v:-\infty \lt v \lt 0\} $
C) $ R_1=\{u:-1 \lt u \lt 1\},R_2=\{v:-\infty \lt v \lt 0\} $
D) $ R_1=\{u:-1\le u\le 1\},R_2=\{v:-\infty \lt v\le 0\} $
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Answer:
Correct Answer: D
Solution:
[d] we have $ fog(x)=f(g(x))=sin(log_{e}| x |). $ $ {\log_{e}}| x | $ has range R, for which $ sin({\log_{e}}| x |)\in [-1,1]. $
Therefore, $ R_1={u:-1\le u\le 1}. $ Also, $ gof(x)=g(f(x))=log_{e}| \sin x |. $
$ \because 0\le | \sin x |\le 1 $ or $ -\infty <{\log_{e}}| \sin x |\le 0 $ Or $ R_2={v:-\infty <v\le 0} $
BETA
