Relations & Functions
Domain and Range of Composite Functions
Grade 12
Question:
<p>If the domain of \(f(x) = \frac{1}{\pi}\cos^{-1}\left[\log_3\left(\frac{x^2}{3}\right)\right]\) where \(x > 0\) is \([a, b]\) and the range of \(f(x)\) is \([c, d]\), then which of the following hold?</p>
<p>(a) \(a, b\) are the roots of the equation \(x^4 - 3x^3 - x + 3 = 0\)</p>
<p>(b) \(a, b\) are the roots of the equation \(x^4 - x^3 + x^2 - 2x + 1 = 0\)</p>
<p>(c) \(a^3 + d^3 = 1\)</p>
<p>(d) \(a^2 + b^2 + c^2 + d^2 = 11\)</p>
Step-by-Step Solution
Key Concept: For f(x) = (1/π)cos⁻¹[log₃(x²/3)] to be defined, the argument of cos⁻¹ must be in [-1, 1]. This constraint determines the domain [a, b], and then we find the range by evaluating f at domain endpoints.
<p><strong>Step 1: Find Domain Constraints</strong></p><p>For f(x) to be defined, we need: -1 ≤ log₃(x²/3) ≤ 1</p><p><strong>Step 2: Solve Left Inequality</strong></p><p>log₃(x²/3) ≥ -1</p><p>x²/3 ≥ 3⁻¹ = 1/3</p><p>x² ≥ 1</p><p>Since x > 0: x ≥ 1, so a = 1</p><p><strong>Step 3: Solve Right Inequality</strong></p><p>log₃(x²/3) ≤ 1</p><p>x²/3 ≤ 3</p><p>x² ≤ 9</p><p>Since x > 0: x ≤ 3, so b = 3</p><p><strong>Domain: [1, 3]</strong></p><p><strong>Step 4: Check Option (a)</strong></p><p>If a=1, b=3 are roots of x⁴ - 3x³ - x + 3 = 0:</p><p>At x=1: 1 - 3 - 1 + 3 = 0 ✓</p><p>At x=3: 81 - 81 - 3 + 3 = 0 ✓</p><p>Option (a) is TRUE</p><p><strong>Step 5: Find Range of f(x)</strong></p><p>At x = 1: log₃(1/3) = -1, so f(1) = (1/π)cos⁻¹(-1) = (1/π)·π = 1</p><p>At x = 3: log₃(3) = 1, so f(3) = (1/π)cos⁻¹(1) = (1/π)·0 = 0</p><p>Since cos⁻¹ is decreasing, f is continuous on [1,3], so Range: [0, 1]</p><p>Thus c = 0, d = 1</p><p><strong>Step 6: Check Option (c)</strong></p><p>a³ + d³ = 1³ + 1³ = 2 ≠ 1, so FALSE</p><p><strong>Step 7: Check Option (d)</strong></p><p>a² + b² + c² + d² = 1² + 3² + 0² + 1² = 1 + 9 + 0 + 1 = 11 ✓</p><p>Option (d) is TRUE</p><p><strong>Step 8: Verify Option (b) is False</strong></p><p>At x=1: 1 - 1 + 1 - 2 + 1 = 0 ✓</p><p>At x=3: 81 - 27 + 9 - 6 + 1 = 58 ≠ 0 ✗</p><p>Option (b) is FALSE</p><p><strong>∴ Answer: a, d</strong></p>
Correct Answer: a, d