The function `f(x)=x/(1+|x|)` is (a) strictly increasing (b) strictly decreasing (c) neither increasing nor decreasing (d) none of these

Explanation

To determine whether the function \( f(x) = \frac{x}{1 + |x|} \) is strictly increasing, strictly decreasing, or neither, we will analyze the function in two cases based on the value of \( x \): when \( x \geq 0 \) and when \( x < 0 \).

### Step 1: Analyze the function for \( x \geq 0 \)

For \( x \geq 0 \), the absolute value function \( |x| \) simplifies to \( x \). Thus, we can rewrite the function as:

\[
f(x) = \frac{x}{1 + x}
...