First slide

Functions (XII)

Question

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

Moderate

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

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

Solution

$f (x) = \sin x a n d g (x) = \log |x|$
$\Rightarrow f o g (x) = f (g (x)) = f (\log |x|) = \sin (\log |x|) \in [- 1, 1] (\sin c e \log |x|, r a n g e i s R)$
$R_{1} = \{u / - 1 \leq u \leq 1\}$
$a l s o, g o f (x) = g (f (x)) = g (\sin x) = \log |\sin x|$
$\sin c e 0 \leq |\sin x| \leq 1 \Rightarrow - \infty < \log |\sin x| \leq 0$
$R_{2} = \{v / - \infty < v \leq 0\}$

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