<p>Let \(f(x) = \frac{1}{1-x}\), then {\(f \circ (f \circ f)\)}(100) is equal to</p>
Step-by-Step Solution
Key Concept: Function composition requires careful substitution of one function into another. We need to find f(f(x)) first, then compose it with f again to get f(f(f(x))), and finally evaluate at x=100. The key is recognizing the cyclic pattern that emerges in repeated compositions.
<p><strong>Step 1:</strong> Find f(f(x)).</p><p>Given: f(x) = 1/(1-x)</p><p>f(f(x)) = f(1/(1-x)) = 1/(1 - 1/(1-x))</p><p>Simplify the denominator: 1 - 1/(1-x) = (1-x-1)/(1-x) = -x/(1-x)</p><p>Therefore: f(f(x)) = 1/(-x/(1-x)) = (1-x)/(-x) = (x-1)/x</p><p><strong>Step 2:</strong> Find f(f(f(x))) = f((x-1)/x).</p><p>f(f(f(x))) = 1/(1 - (x-1)/x)</p><p>Simplify the denominator: 1 - (x-1)/x = (x - (x-1))/x = 1/x</p><p>Therefore: f(f(f(x))) = 1/(1/x) = x</p><p><strong>Step 3:</strong> Interpret the notation {f ∘ (f ∘ f)}(100).</p><p>This means f(f(f(x))), which we found equals x.</p><p><strong>Step 4:</strong> Evaluate at x = 100.</p><p>f(f(f(100))) = 100</p><p><strong>∴ Answer:</strong> 100</p>
Correct Answer: 100