CBSE Class 12 Mathematics Relations and Functions Q #2142
KNOWLEDGE BASED
APPLY
3 Marks Generic-Curriculum SA
Let $D$ be the domain of the real valued function $f$ defined by $f(x) = \sqrt{25 - x^2}$. Then, write $D$.
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Detailed Solution

Step 1: Identify the condition for the domain

For a real-valued function involving a square root, the expression inside the square root must be non-negative. Therefore, we require: $$25 - x^2 \geq 0$$

Step 2: Solve the inequality

Rearranging the inequality, we get: $$x^2 \leq 25$$ Taking the square root on both sides, this is equivalent to: $$|x| \leq 5$$

Step 3: Express the domain in interval notation

The inequality $|x| \leq 5$ implies that $x$ must lie between $-5$ and $5$, inclusive. Thus, the domain $D$ is the closed interval: $$D = [-5, 5]$$

Final Answer: [-5, 5]

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Pedagogical Audit
Bloom's Analysis: This is an APPLY question because the student must apply the mathematical constraint of square root functions to determine the valid input set.
Knowledge Dimension: PROCEDURAL
Justification: The student follows a standard algorithmic procedure to solve a quadratic inequality to find the domain.
Syllabus Audit: In the context of CBSE Class 12, this is classified as KNOWLEDGE. This question tests the fundamental understanding of domain calculation for algebraic functions, which is a core prerequisite for the Calculus unit.

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