Functions
General
Grade 12
Question:
If f_1(x) = 2^{f_2(x)}, where f_2(x) = 2012^{f_3(x)}, where f_3(x) = \left( \frac{1}{2013} \right)^{f^{(x)}}, f_4(x) = \log_{2013}\log_{x}2012, then the range of f_1(x) is -
(A) (2, \infty)
(B) (2012, \infty)
(C) (0, \infty)
(D) (-\infty, \infty)
Step-by-Step Solution
<div class="solution"><p>Let x=n+2/3. The nested expression reduces to [x+2/3]=5, giving n+4/3+2/3=[n+2]=n+2... wait: [x+2/3]=[n+4/3]=n+1=5 â n=4.</p><p>So x=4+2/3=14/3. [x]=4.</p><p><strong>Answer: (A),(D)</strong></p><div class="trap-box"><strong>Trap:</strong> Fractional part stays 2/3 after adding integers. Don't lose track.</div><div class="key-concept"><strong>Key Concept:</strong> Nested [ ] and { } â reduce to x=n+r with 0âĪr<1</div></div>
Correct Answer: A