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# Solution for Prove that the Function F Given by F(X) = X − [X] is Increasing in (0, 1) ? - CBSE (Commerce) Class 12 - Mathematics

ConceptIncreasing and Decreasing Functions

#### Question

Prove that the function f given by f(x) = x − [x] is increasing in (0, 1) ?

#### Solution

$f\left( x \right) = x - \left[ x \right]$

$\text { Let } x_1 , x_2 \in \left( 0, 1 \right) \text { such that } x_1 < x_2 . \text { Then },$

$\left[ x_1 \right]=\left[ x_2 \right]= 0 ...(1)$

$\text { Now,}$

$x_1 < x_2$

$\Rightarrow x_1 - \left[ x_1 \right] < x_2 - \left[ x_2 \right] \left[ \text { From eq }. (1) \right]$

$\Rightarrow f\left( x_1 \right) < f\left( x_2 \right)$

$\therefore x_1 < x_2$

$\Rightarrow f\left( x_1 \right) < f\left( x_2 \right), \forall x_1 , x_2 \in \left( 0, 1 \right)$

$\text { So},f\left( x \right) \text { is increasing on }\left( 0, 1 \right).$

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Solution Prove that the Function F Given by F(X) = X − [X] is Increasing in (0, 1) ? Concept: Increasing and Decreasing Functions.
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