<p>Match each function with its range: (I) <span class="math-inline">\(\frac{\cos^2 x+\cos x+2}{\cos^2 x+\cos x+1}\)</span> (II) trig ratio (III) <span class="math-inline">\(\frac{7}{3(x^6+2x^4+3x^2+1)}\)</span> (IV) <span class="math-inline">\(\log_8(x^2+2x+2)\)</span> with (P)<span class="math-inline">\((0,7/3]\)</span> (Q)<span class="math-inline">\([4/3,7/3]\)</span> (R)<span class="math-inline">\([0,1/3]\)</span> (S)<span class="math-inline">\([0,\infty)\)</span></p>
Step-by-Step Solution
Key Concept: General
<div class="solution"><p>(I): Let t=cos x∈[-1,1]. t²+t+1∈[3/4,3]. Expression=1+1/(t²+t+1)∈[4/3,7/3] → (Q)</p><p>(II): Simplifies to (cos x-sin x)/√(3(2-u²)) where u=cos x-sin x. Range [0,1/3] → (R)</p><p>(III): x⁶+2x⁴+3x²+1≥1, so expression∈(0,7/3] → (P)</p><p>(IV): (x+1)²+1≥1, so log₈≥0, unbounded → (S)</p><p><strong>Answer: I→Q, II→R, III→P, IV→S</strong></p><div class="trap-box"><strong>Trap:</strong> In (II), domain restrictions on sin x and cos x are critical.</div><div class="key-concept"><strong>Key Concept:</strong> Match-list range problems — reduce each to one-variable range problem</div></div>
Correct Answer: I→Q, II→R, III→P, IV→S