Q1421: Find the domain of the function $f(x)=\cos^{-1}(x^{2}-4)$....

Class CBSE Class 12 Mathematics Inverse Trigonometric Functions Q #1421

KNOWLEDGE BASED

UNDERSTAND

2 Marks 2025 AISSCE(Board Exam) VSA

Find the domain of the function $f(x)=\cos^{-1}(x^{2}-4)$.

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AI Tutor Explanation

Detailed Solution

Step 1: Understanding the Domain of Inverse Cosine Function

The domain of the inverse cosine function, $\cos^{-1}(u)$, is $-1 \le u \le 1$. This means that the input to the $\cos^{-1}$ function must be between -1 and 1, inclusive.

Step 2: Applying the Domain Condition to the Given Function

For the function $f(x) = \cos^{-1}(x^2 - 4)$, we must have $-1 \le x^2 - 4 \le 1$.

Step 3: Solving the Inequality

We can split this compound inequality into two separate inequalities:\r\n\r\n1. $x^2 - 4 \le 1$ which implies $x^2 \le 5$, so $-\sqrt{5} \le x \le \sqrt{5}$.\r\n2. $x^2 - 4 \ge -1$ which implies $x^2 \ge 3$, so $x \le -\sqrt{3}$ or $x \ge \sqrt{3}$.

Step 4: Combining the Inequalities

We need to find the intersection of the two solution sets:\r\n\r\n$-\sqrt{5} \le x \le \sqrt{5}$ and ($x \le -\sqrt{3}$ or $x \ge \sqrt{3}$).\r\n\r\nThis gives us the intervals $[-\sqrt{5}, -\sqrt{3}]$ and $[\sqrt{3}, \sqrt{5}]$.

Step 5: Expressing the Domain

Therefore, the domain of the function $f(x) = \cos^{-1}(x^2 - 4)$ is $[-\sqrt{5}, -\sqrt{3}] \cup [\sqrt{3}, \sqrt{5}]$.

Final Answer: $[-\sqrt{5}, -\sqrt{3}] \cup [\sqrt{3}, \sqrt{5}]$

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Pedagogical Audit

Bloom's Analysis: This is an UNDERSTAND question because the student needs to understand the domain of the inverse cosine function and apply that understanding to find the domain of the given function.

Knowledge Dimension: CONCEPTUAL

Justification: The question requires understanding the concept of the domain of a function, specifically the inverse cosine function, and how transformations affect the domain.<\/span>

Syllabus Audit: In the context of CBSE Class 12, this is classified as KNOWLEDGE. The question directly tests the student's knowledge of inverse trigonometric functions and their domains, a standard topic in the syllabus.