Question

The modulus function $f: \mathbb{R} \to \mathbb{R}^{+}$ given by $f(x) = |x|$ will be:

• A. one-one and onto
• B. many-one and onto
• C. one-one and into
• D. many-one and into

Hint:

The modulus function $f(x) = |x|$ is many-one because $f(a) = f(-a)$, and it is onto $\mathbb{R}^{+}$ since every non-negative real number has a preimage.

Answer 1

The Correct Option is B

Step 1: Understanding the modulus function.
The modulus function $f(x) = |x|$ maps every real number $x$ to a non-negative real number. Its codomain is $\mathbb{R}^{+}$ (non-negative real numbers).

Step 2: Check one-one property.
If $f(a) = f(b)$, then $|a| = |b|$. This means either $a = b$ or $a = -b$. Thus, different inputs (e.g., $2$ and $-2$) give the same output. Hence, $f(x)$ is **not one-one**, but **many-one**.

Step 3: Check onto property.
For every $y \in \mathbb{R}^{+}$, there exists an $x \in \mathbb{R}$ such that $f(x) = |x| = y$. Thus, the function is **onto** $\mathbb{R}^{+}$.

Step 4: Conclusion.
The function is **many-one and onto**, so the correct answer is (B).