<p>Given \( f_1(x) = \dfrac{1}{x} \), \( f_2(x) = 1 - x \), \( f_3(x) = \dfrac{1}{1-x} \) and \( (f_2 \circ J \circ f_1)(x) = f_3(x) \). Then \( J(x) \) equals:</p>
Step-by-Step Solution
Key Concept: To find J(x), decompose the composition equation by applying inverse operations: isolate J by computing f₂⁻¹ and f₁⁻¹, then substitute to get J(x) = f₂⁻¹(f₃(f₁(x))).
<p><strong>Step 1:</strong> Given (f₂ ∘ J ∘ f₁)(x) = f₃(x), which means f₂(J(f₁(x))) = f₃(x).</p><p><strong>Step 2:</strong> Apply f₂⁻¹ to both sides. Since f₂(x) = 1 - x, we have f₂⁻¹(x) = 1 - x (self-inverse).<br/>Therefore: J(f₁(x)) = f₂⁻¹(f₃(x)) = 1 - f₃(x) = 1 - 1/(1-x) = (1-x-1)/(1-x) = -x/(1-x)</p><p><strong>Step 3:</strong> Let u = f₁(x) = 1/x, so x = 1/u. Substitute into J(f₁(x)):<br/>J(u) = -((1/u)/(1 - 1/u)) = -(1/u)/((u-1)/u) = -1/(u-1) = 1/(1-u)</p><p><strong>Step 4:</strong> Replace u with x: J(x) = 1/(1-x) = f₃(x)</p><p>∴ Answer: <strong>C</strong> (J(x) = 1/(1-x))</p>
Correct Answer: C