Q1267: Find the domain of the function $f(x)=sin^{-1}(x^{2}-4).$ Al...

Class CBSE Class 12 Mathematics Inverse Trigonometric Functions Q #1267

KNOWLEDGE BASED

APPLY

2 Marks 2024 AISSCE(Board Exam) VSA

Find the domain of the function $f(x)=sin^{-1}(x^{2}-4).$ Also, find its range.

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AI Tutor Explanation

Detailed Solution

Step 1: Understanding the Domain of Inverse Sine Function

The domain of the inverse sine function, $sin^{-1}(u)$, is $-1 \le u \le 1$. Therefore, for $f(x) = sin^{-1}(x^2 - 4)$ to be defined, we must have $-1 \le x^2 - 4 \le 1$.

Step 2: Solving the Inequality

We have two inequalities to solve: 1. $x^2 - 4 \ge -1$ 2. $x^2 - 4 \le 1$

Step 3: Solving the First Inequality

$x^2 - 4 \ge -1$ $x^2 \ge 3$ This implies $x \le -\sqrt{3}$ or $x \ge \sqrt{3}$.

Step 4: Solving the Second Inequality

$x^2 - 4 \le 1$ $x^2 \le 5$ This implies $-\sqrt{5} \le x \le \sqrt{5}$.

Step 5: Combining the Inequalities to Find the Domain

We need to find the intersection of the two solution sets: $x \le -\sqrt{3}$ or $x \ge \sqrt{3}$ and $-\sqrt{5} \le x \le \sqrt{5}$. This gives us the domain: $[-\sqrt{5}, -\sqrt{3}] \cup [\sqrt{3}, \sqrt{5}]$.

Step 6: Finding the Range

To find the range, we need to determine the possible values of $f(x) = sin^{-1}(x^2 - 4)$ for $x$ in the domain $[-\sqrt{5}, -\sqrt{3}] \cup [\sqrt{3}, \sqrt{5}]$. When $x = \pm \sqrt{3}$, $x^2 - 4 = 3 - 4 = -1$. So, $sin^{-1}(-1) = -\frac{\pi}{2}$. When $x = \pm \sqrt{5}$, $x^2 - 4 = 5 - 4 = 1$. So, $sin^{-1}(1) = \frac{\pi}{2}$. Since $x^2 - 4$ varies continuously from -1 to 1 as $x$ varies from $\sqrt{3}$ to $\sqrt{5}$ (and from $-\sqrt{5}$ to $-\sqrt{3}$), the range of $f(x)$ is $[-\frac{\pi}{2}, \frac{\pi}{2}]$.

Final Answer: Domain: $[-\sqrt{5}, -\sqrt{3}] \cup [\sqrt{3}, \sqrt{5}]$; Range: $[-\frac{\pi}{2}, \frac{\pi}{2}]$

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Pedagogical Audit

Bloom's Analysis: This is an APPLY question because the student needs to apply their knowledge of inverse trigonometric functions and inequalities to find the domain and range of the given function.

Knowledge Dimension: PROCEDURAL

Justification: The student needs to follow a specific procedure to find the domain and range, involving solving inequalities and understanding the properties of the inverse sine function.

Syllabus Audit: In the context of CBSE Class 12, this is classified as KNOWLEDGE. The question directly tests the student's understanding of inverse trigonometric functions, which is a core concept in the syllabus.