Q571: A function $f:\mathbb{R}\rightarrow\mathbb{R}$ defined as ...

Class CBSE Class 12 Mathematics Relations and Functions Q #571

KNOWLEDGE BASED

APPLY

1 Marks 2024 AISSCE(Board Exam) MCQ SINGLE

A function $f:\mathbb{R}\rightarrow\mathbb{R}$ defined as $f(x)=x^{2}-4x+5$ is:

(A) injective but not surjective.

(B) surjective but not injective.

(D) neither injective nor surjective.

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Correct Answer: D

AI Tutor Explanation

Step-by-Step Solution

Step 1: Analyze the function for injectivity.

A function $f(x)$ is injective (one-to-one) if for every $x_1, x_2$ in the domain, $f(x_1) = f(x_2)$ implies $x_1 = x_2$. Equivalently, if $x_1 \neq x_2$, then $f(x_1) \neq f(x_2)$.

Consider $f(x) = x^2 - 4x + 5$. We can rewrite this as $f(x) = (x-2)^2 + 1$. This is a parabola with vertex at $(2, 1)$ opening upwards.

For example, $f(1) = (1-2)^2 + 1 = 2$ and $f(3) = (3-2)^2 + 1 = 2$. Since $f(1) = f(3)$ but $1 \neq 3$, the function is not injective.

Step 2: Analyze the function for surjectivity.

A function $f(x)$ is surjective (onto) if for every $y$ in the codomain, there exists an $x$ in the domain such that $f(x) = y$. In other words, the range of the function is equal to the codomain.

Since $f(x) = (x-2)^2 + 1$, the minimum value of $f(x)$ is 1 (when $x = 2$). Therefore, the range of $f(x)$ is $[1, \infty)$.

The codomain is $\mathbb{R}$, which is $(-\infty, \infty)$. Since the range $[1, \infty)$ is not equal to the codomain $(-\infty, \infty)$, the function is not surjective.

Correct Answer: neither injective nor surjective.<\/strong>

AI Suggestion: Option D

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Pedagogical Audit

Bloom's Analysis: This is an APPLY question because the student needs to apply their understanding of injective and surjective functions to the given quadratic function to determine its nature.

Knowledge Dimension: CONCEPTUAL
Justification: The question requires understanding the concepts of injective and surjective functions and how they relate to the properties of a given function.

Syllabus Audit: In the context of CBSE Class 12, this is classified as KNOWLEDGE. The question directly tests the student's understanding of the definitions and properties of injective and surjective functions, which is a core concept covered in the textbook.

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