Question

The mapping $ f: \mathbb{R} \to \mathbb{R} $ such that $ f(x) = |x - 1| $, $ x \in \mathbb{R} $ is:

• A. one-one, onto
• B. many-one, onto
• C. one-one, into
• D. neither one-one nor onto

Hint:

A function involving absolute values is often not one-one due to symmetry, and it may not be onto if it does not cover the entire codomain.

Answer 1

The Correct Option is D

The function \( f(x) = |x - 1| \) is a piecewise function: \[ f(x) = \begin{cases} x - 1, & x \geq 1 1 - x, & x<1 \end{cases} \]
Step 1: Check if the function is one-one.
For a function to be one-one, it must not take the same value for different inputs. However, for \( x = 0 \) and \( x = 2 \), both give \( f(0) = f(2) = 1 \). Therefore, the function is not one-one.
Step 2: Check if the function is onto.
For a function to be onto, every element in the codomain (in this case, \( \mathbb{R} \)) must have a corresponding element in the domain. The function \( f(x) = |x - 1| \) only takes non-negative values, so it cannot cover all of \( \mathbb{R} \). Therefore, the function is not onto. Hence, the function is neither one-one nor onto.