Relations & Functions
Range of functions
Grade 12
Question:
<p>If \(f(x) = \dfrac{3}{1 + \tan^2 x} + \dfrac{9}{1 + \cot^2 x}\), then:</p>
<p>(a) number of integers in the range of \(f(x)\) is 7.</p>
<p>(b) number of integers in the range of \(f(x)\) is 5.</p>
<p>(c) sum of the integers in the range of \(f(x)\) is 30.</p>
<p>(d) sum of the integers in the range of \(f(x)\) is 42.</p>
Step-by-Step Solution
Key Concept: Use the identities 1 + tan²x = sec²x and 1 + cot²x = csc²x to convert the expression into trigonometric functions, then simplify using cos²x + sin²x = 1.
<p><strong>Step 1:</strong> Apply Pythagorean identities</p><p>• 1 + tan²x = sec²x = 1/cos²x</p><p>• 1 + cot²x = csc²x = 1/sin²x</p><p><strong>Step 2:</strong> Rewrite the function</p><p>f(x) = 3/(1/cos²x) + 9/(1/sin²x)</p><p>f(x) = 3cos²x + 9sin²x</p><p><strong>Step 3:</strong> Factor and simplify</p><p>f(x) = 3cos²x + 9sin²x</p><p>f(x) = 3cos²x + 3sin²x + 6sin²x</p><p>f(x) = 3(cos²x + sin²x) + 6sin²x</p><p>f(x) = 3(1) + 6sin²x</p><p>f(x) = 3 + 6sin²x</p><p><strong>Step 4:</strong> Determine range</p><p>Since 0 ≤ sin²x ≤ 1:</p><p>• Minimum value: f(x) = 3 + 6(0) = 3</p><p>• Maximum value: f(x) = 3 + 6(1) = 9</p><p>∴ Range is [3, 9] and f(x) is periodic with period π</p>
Correct Answer: A,D