Class CBSE Class 12 Mathematics Relations and Functions Q #573
KNOWLEDGE BASED
APPLY
1 Marks 2024 AISSCE(Board Exam) MCQ SINGLE
Let \(f:R_{+}\rightarrow[-5,\infty)\) be defined as \(f(x)=9x^{2}+6x-5\), where \(R_{+}\) is the set of all non-negative real numbers. Then, f is:
(A) one-one
(B) onto
(C) bijective
(D) neither one-one nor onto
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AI Tutor Explanation

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Step-by-Step Solution

  1. Check if the function is one-one:<\/strong><\/p>

    Given \(f(x) = 9x^2 + 6x - 5\). To check if it's one-one, we can analyze its derivative.

    \(f'(x) = 18x + 6\). Since \(x \in R_{+}\), \(x \geq 0\), so \(f'(x) \geq 6 > 0\). This means the function is strictly increasing for \(x \geq 0\). Therefore, it is one-one.

  2. Check if the function is onto:<\/strong><\/p>

    The range of the function is given as \([-5, \infty)\). We need to check if the function actually attains all values in this range.

    We can rewrite \(f(x)\) as \(f(x) = (3x + 1)^2 - 6\). Since \(x \in R_{+}\), the minimum value of \(3x + 1\) is 1 (when x=0). Therefore, the minimum value of \((3x + 1)^2\) is 1, and the minimum value of \(f(x)\) is \(1 - 6 = -5\).

    As \(x\) increases from 0, \((3x + 1)^2\) increases, and so does \(f(x)\). Since \(x\) can take any non-negative real value, \(f(x)\) can take any value greater than or equal to -5. Thus, the range of \(f(x)\) is indeed \([-5, \infty)\), which matches the given codomain. Therefore, the function is onto.

  3. Conclusion:<\/strong><\/p>

    Since the function is both one-one and onto, it is bijective.

Correct Answer: bijective<\/strong>

AI Suggestion: Option C

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Pedagogical Audit
Bloom's Analysis: This is an APPLY question because it requires students to apply their understanding of functions, specifically one-one and onto functions, to a given function and determine its properties. They need to use the definition of these function types and apply algebraic manipulation to reach the solution.
Knowledge Dimension: CONCEPTUAL
Justification: The question tests the understanding of concepts related to functions, such as one-one (injective) and onto (surjective) functions. It requires the student to understand the definitions and properties of these types of functions and apply them to the given function.
Syllabus Audit: In the context of CBSE Class 12, this is classified as KNOWLEDGE. The question directly assesses the student's understanding of the definitions and properties of one-one and onto functions, which are core concepts covered in the Relations and Functions chapter of the textbook.

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