Sets, Relations & Functions
General
Grade 11
Question:
For \( x \in \mathbb{R} - \{0, 1\} \), let f_1(x) = \frac{1}{x}, f_2(x) = 1 - x and f_3(x) = \frac{1}{1 - x} \) be three given functions. If a function, J(x) satisfies \( f_2 \circ f_1 \circ f(x) = f_3(x) \) then J(x) is equal to :-
1. f_3(x)
2. f_1(x)
3. f_2(x)
4. \frac{1}{x} f_3(x)
Step-by-Step Solution
<div class="solution"><p>Need: 4-x²≠0 → x≠±2; and x³-x>0 → x(x-1)(x+1)>0 → x∈(-1,0)∪(1,∞). Exclude x=2: domain=(-1,0)∪(1,2)∪(2,∞).</p><div class="trap-box"><strong>Trap:</strong> Logarithm needs strict positivity.</div><div class="key-concept"><strong>Key Concept:</strong> Domain = intersection of domains of all component pieces</div></div>
Correct Answer: 1