Sets, Relations & Functions
General
Grade 11
Question:
<p>For <span class="math-inline">\(x\in\mathbb{R}\setminus\{0,1\}\)</span>, <span class="math-inline">\(f_1(x)=1/x, f_2(x)=1-x, f_3(x)=1/(1-x)\)</span>. If <span class="math-inline">\((f_2\circ I\circ f_1)(x)=f_3(x)\)</span>, find <span class="math-inline">\(I(x)\)</span>.</p>
Step-by-Step Solution
Key Concept: General
<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: I(x) = 1/(1-x)