Show that function `f: R ->{x in R : -1 < x < 1}` defined by `f(x)=x/(1+|x|), x in R` is one one and onto function

Explanation

To show that the function \( f: \mathbb{R} \to (-1, 1) \) defined by \( f(x) = \frac{x}{1 + |x|} \) is one-to-one and onto, we will proceed in two parts: proving that it is one-to-one and then proving that it is onto.

### Step 1: Proving that \( f \) is one-to-one

To prove that \( f \) is one-to-one, we need to show that if \( f(x_1) = f(x_2) \), then \( x_1 = x_2 \).

#### Case 1: Both \( x_1 \) and \( x_2 \) are non-negative (i.e., \( x_1, x_2 \geq 0 \))

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