Relations & Functions
Domain and Range
Grade 12
Question:
<p>The domain of the definition of the function <i>f</i>(<i>x</i>) = <span style="text-decoration: overline;"><i>1</i>/(4 − <i>x</i><sup>2</sup>)</span> + log₁₀(<i>x</i><sup>3</sup> − <i>x</i>) is</p>
<p>(a) (−1, 0) ∪ (1, 2) ∪ (3, ∞)</p>
<p>(b) (−2, −1) ∪ (−1, 0) ∪ (2, ∞)</p>
<p>(c) (−1, 0) ∪ (1, 2) ∪ (2, ∞)</p>
<p>(d) (1, 2) ∪ (2, ∞)</p>
Step-by-Step Solution
Key Concept: The domain requires three conditions: the expression under the square root must be positive, the argument of the logarithm must be positive, and the denominator must be non-zero. Find each condition separately, then take their intersection.
<p><strong>Step 1:</strong> For the square root term, we need <i>1</i>/(4 − <i>x</i><sup>2</sup>) ≥ 0.</p><p>This requires 4 − <i>x</i><sup>2</sup> > 0, so <i>x</i><sup>2</sup> < 4, giving −2 < <i>x</i> < 2.</p><p><strong>Step 2:</strong> For the logarithm term, we need <i>x</i><sup>3</sup> − <i>x</i> > 0.</p><p><i>x</i>(<i>x</i><sup>2</sup> − 1) > 0</p><p><i>x</i>(<i>x</i> − 1)(<i>x</i> + 1) > 0</p><p><strong>Step 3:</strong> Using the wavy curve method for <i>x</i>(<i>x</i> − 1)(<i>x</i> + 1) > 0:</p><p>The expression is positive on (−1, 0) ∪ (1, ∞).</p><p><strong>Step 4:</strong> Also, 4 − <i>x</i><sup>2</sup> ≠ 0, so <i>x</i> ≠ ±2.</p><p><strong>Step 5:</strong> Taking the intersection of all constraints:</p><p>(−2, 2) ∩ [(−1, 0) ∪ (1, ∞)] ∩ [<i>x</i> ≠ ±2] = (−1, 0) ∪ (1, 2) ∪ (2, ∞).</p><p>∴ Answer is (c).</p>
Correct Answer: C