If f:R→R and g:R→R are defined by fx=x and gx=x−3 ∀x∈R,then gfx:−85<x<85=

gfx=gx=x−3=8 x−3if −85<x<85⇒ 0<x<1.6if 0<x<1 then x=0if 1<x<1.6 then x=1∴ gfx=0−3, 1−3=−3, −2∴ Range =−3, −2

If f:R→R and g:R→R are defined by fx=x and gx=x−3 ∀x∈R,then gfx:−85<x<85=

gfx=gx=x−3=8 x−3if −85<x<85⇒ 0<x<1.6if 0<x<1 then x=0if 1<x<1.6 then x=1∴ gfx=0−3, 1−3=−3, −2∴ Range =−3, −2

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$\begin{array}{l} I f f : R \to R a n d g : R \to R a r e d e f i n e d b y f (x) = |x| a n d g (x) = [x - 3] \forall x \in R, \\ t h e n \{g (f (x)) : - \frac{8}{5} < x < \frac{8}{5}\} = \end{array}$

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$\{0, 1\}$

$\{1, 2\}$

$\{- 3, - 2\}$

$\{2, 3\}$

answer is C.

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Detailed Solution

$\begin{array}{l} g (f (x)) = g (|x|) = [|x| - 3] = 8 [|x|] - 3 \\ i f - \frac{8}{5} < x < \frac{8}{5} \\ \Rightarrow 0 < |x| < 1.6 \\ i f 0 < |x| < 1 t h e n [|x|] = 0 \\ i f 1 < |x| < 1.6 t h e n [|x|] = 1 \\ ∴ g (f (x)) = 0 - 3, 1 - 3 \\ = - 3, - 2 \\ ∴ R a n g e = \{- 3, - 2\} \end{array}$