Question

The function $f: \mathbb{R} \to \mathbb{R}$ defined by $f(x) = x^{2} + 5$ will be:

• A. one-one and onto
• B. many-one and onto
• C. one-one, but not onto
• D. neither one-one nor onto

Hint:

For quadratic functions like $f(x) = x^{2} + c$, the function is many-one and not onto $\mathbb{R}$ (its range is restricted).

Answer 1

The Correct Option is D

Step 1: Check one-one property.
A function is one-one if $f(a) = f(b) \implies a = b$. Here, $f(x) = x^{2} + 5$. Take $a=2, b=-2$: \[ f(2) = 2^{2} + 5 = 9, f(-2) = (-2)^{2} + 5 = 9 \] So, $f(2) = f(-2)$ but $2 \neq -2$. Hence, the function is **not one-one (many-one)**.

Step 2: Check onto property.
The codomain is $\mathbb{R}$, but the range is $[5, \infty)$. For example, there is no $x \in \mathbb{R}$ such that $f(x) = 3$. Therefore, the function is **not onto**.

Step 3: Conclusion.
Since the function is neither one-one nor onto, the correct answer is (D).