#### Question

The function \[f\left( x \right) = \frac{x}{1 + \left| x \right|}\] is

(a) strictly increasing

(b) strictly decreasing

(c) neither increasing nor decreasing

(d) none of these

#### Solution

(a) strictly increasing

\[f\left( x \right) = \frac{x}{1 + \left| x \right|}\]

\[\text { Case 1: When }x > 0, \left| x \right| = x\]

\[f\left( x \right) = \frac{x}{1 + \left| x \right|}\]

\[ = \frac{x}{1 + x}\]

\[ \Rightarrow f'\left( x \right) = \frac{\left( 1 + x \right)1 - x\left( 1 \right)}{\left( 1 + x \right)^2}\]

\[ = \frac{1}{\left( 1 + x \right)^2} > 0, \forall x \in R\]

\[\text { So,f }\left( x \right) \text { is strictly increasing when }x> 0.\]

\[\text { Case 2: When }x < 0, \left| x \right| = - x\]

\[f\left( x \right) = \frac{x}{1 + \left| x \right|}\]

\[ = \frac{x}{1 - x}\]

\[ \Rightarrow f'\left( x \right) = \frac{\left( 1 - x \right)1 - x\left( - 1 \right)}{\left( 1 - x \right)^2}\]

\[ = \frac{1}{\left( 1 - x \right)^2} > 0, \forall x \in R\]

\[\text { So,f }\left( x \right) \text { is strictly increasing when }x <0.\]

\[\text { Thus,f }\left( x \right) \text { is strictly increasing on R } . \]