Solution:

It is given that, $ f: \mathbf{R} \rightarrow\{x \in \mathbf{R}:-1\lt x \lt1\} \\ $

defined by, $ f(x)=\frac{x}{1+|x|}, x \in \mathbf{R} \\ $

Suppose f(x) = f(y), where x, y ∈ R.

$$ \Rightarrow \frac{x}{1+|x|}=\frac{y}{1+|y|} \\ $$

It can be observed that if x is positive and y is …