Relations & Functions
Domain and Range
Grade 12
Question:
<p>Find the domain of the function <span style='font-family:monospace'>f(x) = log₃ log₁/₃(x² + 10x + 25) + 1/([x] + 5)</span> (where <span style='font-family:monospace'>[·]</span> denotes the greatest integer function).</p>
<p>(a) <span style='font-family:monospace'>(-4, -3)</span></p>
<p>(b) <span style='font-family:monospace'>(-6, -5)</span></p>
<p>(c) <span style='font-family:monospace'>(-6, -4)</span></p>
<p>(d) None of these</p>
Step-by-Step Solution
Key Concept: Domain requires argument of logarithm > 0, and since base is 1/3 < 1, logarithm reverses the inequality. Also handle the greatest integer function in denominator.
<p><strong>Step 1:</strong> For <span style='font-family:monospace'>log₃ log₁/₃(x² + 10x + 25)</span> to be defined:</p><p><span style='font-family:monospace'>x² + 10x + 25 ≠ 0</span>, i.e., <span style='font-family:monospace'>(x + 5)² ≠ 0</span> ⟹ <span style='font-family:monospace'>x ≠ -5</span> ... (i)</p><p><strong>Step 2:</strong> Also, <span style='font-family:monospace'>log₁/₃(x² + 10x + 25) > 0</span></p><p>⟹ <span style='font-family:monospace'>x² + 10x + 25 < 1</span></p><p>⟹ <span style='font-family:monospace'>x² + 10x + 24 < 0</span></p><p>⟹ <span style='font-family:monospace'>(x + 6)(x + 4) < 0</span></p><p>⟹ <span style='font-family:monospace'>x ∈ (-6, -4)</span> ... (ii)</p><p><strong>Step 3:</strong> For the denominator <span style='font-family:monospace'>[x] + 5 ≠ 0</span>, we need <span style='font-family:monospace'>[x] ≠ -5</span>.</p><p>Combining (i) and (ii), the domain is <span style='font-family:monospace'>(-6, -4)</span>.</p><p>∴ Answer is (c).</p>
Correct Answer: c