Sets, Relations & Functions
General
Grade 11
Question:
If a \( \in \mathbb{R} \) and the equation \( -3(x - \lfloor x \rfloor)^2 + 2(x - \lfloor x \rfloor) + a^2 = 0 \) (where \( \lfloor x \rfloor \) denotes the greatest integer \( \leq x \)) has non integral real solution, then all possible values of \( a \) lie in the interval :
(1) \( (-1, 0) \cup (0, 1) \)
(2) \( (1, 2) \)
(3) \( (2, -1) \)
(4) \( (-\infty, -2) \cup (2, \infty) \)
Step-by-Step Solution
<div class="solution"><p>Undo fâ: 1-I(1/x)=1/(1-x) â I(1/x)=x/(x-1)=1/(1-1/x). Let t=1/x: I(t)=1/(1-t).</p><p><strong>Answer: <span class="math-inline">\(I(x)=\dfrac{1}{1-x}\)</span></strong></p><div class="trap-box"><strong>Trap:</strong> Don't confuse I with identity function. Strip outer functions one by one.</div><div class="key-concept"><strong>Key Concept:</strong> Composition equation â peel outer functions to isolate unknown</div></div>
Correct Answer: 1