Sets, Relations & Functions
Domain and Range
Grade 11
Question:
<p>Find the domain of the function <span>f(x) = 1/(4x - |x<sup>2</sup> - 10x + 9|)</span></p>
<p>(a) <span>(7 - √40, 7 + √40)</span></p>
<p>(b) <span>(0, 7 + √40)</span></p>
<p>(c) <span>(7 - √40, ∞)</span></p>
<p>(d) None of these</p>
Step-by-Step Solution
Key Concept: To find the domain, ensure the denominator is non-zero. Handle the absolute value by considering cases based on the sign of the expression inside.
<p><strong>Solution:</strong></p><p>The function <span>f(x) = 1/(4x - |x<sup>2</sup> - 10x + 9|)</span> exists if <span>4x - |x<sup>2</sup> - 10x + 9| ≠ 0</span></p><p>i.e., <span>|x<sup>2</sup> - 10x + 9| < 4x</span></p><p>Note that <span>x<sup>2</sup> - 10x + 9 = (x-1)(x-9)</span></p><p><strong>Case I:</strong> When <span>x ≤ 1</span> or <span>x ≥ 9</span>:</p><p><span>x<sup>2</sup> - 10x + 9 < 4x</span></p><p><span>x<sup>2</sup> - 14x + 9 < 0</span></p><p><span>(x - 7)² < 40</span></p><p><span>x ∈ (7 - √40, 7 + √40)</span> [but x ≤ 1 or x ≥ 9]</p><p>∴ <span>x ∈ (7 - √40, 1] ∪ [9, 7 + √40)</span></p><p><strong>Case II:</strong> When <span>1 < x < 9</span>:</p><p><span>-(x<sup>2</sup> - 10x + 9) < 4x</span></p><p><span>-x<sup>2</sup> + 10x - 9 < 4x</span></p><p><span>-x<sup>2</sup> + 6x - 9 < 0</span></p><p><span>-(x - 3)² < 0</span> (always true except at x = 3)</p><p>∴ <span>x ∈ (1, 9) - {3}</span></p><p><strong>Combined:</strong> <span>x ∈ (7 - √40, 1] ∪ (1, 9) - {3} ∪ [9, 7 + √40) = (7 - √40, 7 + √40) - {3}</span></p><p>The answer is <strong>(a)</strong></p>
Correct Answer: A