Question: A function $f: \mathbb{R} \to \mathbb{R}$ defined as $f(x) = x^{2} - 4x + 5$ is :
Step 1 (Definition of Injective and Surjective): [1/3 Mark]
- A function $f$ is injective (one-one) if $f(x_1) = f(x_2) \implies x_1 = x_2$.
- A function $f$ is surjective (onto) if the Range of $f$ is equal to the Codomain ($\mathbb{R}$).
Step 2 (Step-by-Step Mathematical Working): [1/3 Mark]
Check for Injective:
$$f(x) = x^2 - 4x + 5 = (x-2)^2 + 1$$Consider $f(1) = (1-2)^2 + 1 = 2$ and $f(3) = (3-2)^2 + 1 = 2$.
Since $f(1) = f(3)$ but $1 \neq 3$, the function is not injective.
Check for Surjective:
Since $(x-2)^2 \geq 0$ for all $x \in \mathbb{R}$, it follows that $(x-2)^2 + 1 \geq 1$.
The Range of $f$ is $[1, \infty)$. Since the Codomain is $\mathbb{R}$ and $[1, \infty) \neq \mathbb{R}$, the function is not surjective.
Step 3 (Final Concluded Answer): [1/3 Mark]
Since the function is neither injective nor surjective, the correct option is (D).
💡 EL10 Concept Insight: Think of this function as a parabola that opens upwards starting from a height of 1; because it's a U-shape, it hits the same height twice (not one-to-one), and because it never goes below 1, it misses all the negative numbers in the codomain (not onto).