Sets, Relations & Functions
Grade 11
Question:
<p>For \(x\in\mathbb{R}\setminus\{0,1\}\), \(f_1(x)=1/x, f_2(x)=1-x, f_3(x)=1/(1-x)\). If \((f_2\circ I\circ f_1)(x)=f_3(x)\), find \(I(x)\).</p>
Step-by-Step Solution
<div class="solution"><p>Undo f_2: 1-I(1/x)=1/(1-x) \to I(1/x)=x/(x-1)=1/(1-1/x). Let t=1/x: I(t)=1/(1-t).</p><p><strong>Answer: $I(x)=\dfrac{1}{1-x}$</strong></p><div class="trap-box"><strong>Trap:</strong> Don't confuse I with identity function. Strip outer functions one by one.<div class="key-concept"><strong>Key Concept:</strong> Composition equation -- peel outer functions to isolate unknown
Correct Answer: I(x) = 1/(1-x)