Show that the Modulus Function f : R → R, given by f(x) = |x|, is neither oneone nor onto, where | x | is x, if x is positive or 0 and |x| is – x, if x is negative.

Explanation

To show that the modulus function \( f : \mathbb{R} \to \mathbb{R} \) given by \( f(x) = |x| \) is neither one-one nor onto, we will analyze both properties step by step.

### Step 1: Show that the function is not one-one (injective)

A function is one-one (or injective) if different inputs map to different outputs. In other words, if \( f(a) = f(b) \), then it must follow that \( a = b \).

1. **Choose values for \( x \)**:
- Let \( x_1 = 1 \) and \( x_2 = -1 \).
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