First slide

Functions (XII)

Question

$If f (x) = 2 x + | x |, g (x) = \frac{1}{3} (2 x - | x |) and h (x) = f (g (x)) then domain of \sin^{- 1} \underset{n times}{\underset{⏟}{(h (h (h (h \dots h (x) \dots))))}} is$

Moderate

A

$[- 1, 1]$
B

$[- 1, - \frac{1}{2}] \cup [\frac{1}{2}, 1]$
C

$[- 1, - \frac{1}{2}]$
D

$[\frac{1}{2}, 1]$

Solution

$f (x) = 2 x + x, x \geq 0 = 2 x - x, x < 0$
then $f (x) = 3 x, x \geq 0 = x, x < 0$
$g (x) = \frac{1}{3} (2 x - x), i f x \geq 0 = \frac{1}{3} (2 x + x), x < 0$ $\Rightarrow g (x) = \frac{x}{3}, x \geq 0 = x, x < 0$
$t h e r e f o r e f (g (x)) = 3 (\frac{x}{3}) = x, i f x \geq 0 = x, i f x < 0$ $\Rightarrow h (x) = x, f o r a l l x \in R$
$\sin^{- 1} (h (h (h . . . . . . h (x) . . . . .))) = \sin^{- 1} (x), i t' s d o m a i n i s = [- 1, 1]$

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