Sets, Relations & Functions
Composition of Functions
Grade 11
Question:
<p>For \(x \in R - \{0, 1\}\), let \(f_1(x) = \dfrac{1}{x}\), \(f_2(x) = 1 - x\) and \(f_3(x) = \dfrac{1}{1-x}\) be three given functions. If a function, \(J(x)\) satisfies \((f_2 \circ J \circ f_1)(x) = f_3(x)\) then \(J(x)\) is equal to:</p>
<p>\(f_3(x)\)</p>
<p>\(\dfrac{1}{x} f_3(x)\)</p>
<p>\(f_2(x)\)</p>
<p>\(f_1(x)\)</p>
Step-by-Step Solution
Key Concept: Work backwards through the composition by substituting and then solving for J(x): if (f₂ ∘ J ∘ f₁)(x) = f₃(x), then f₂(J(f₁(x))) = f₃(x), so find what J must be by reversing the operations.
<p><strong>Step 1:</strong> Given: (f₂ ∘ J ∘ f₁)(x) = f₃(x)</p><p>This means: f₂(J(f₁(x))) = f₃(x)</p><p><strong>Step 2:</strong> Substitute f₁(x) = 1/x and f₃(x) = 1/(1-x):</p><p>f₂(J(1/x)) = 1/(1-x)</p><p><strong>Step 3:</strong> Since f₂(y) = 1 - y, we have:</p><p>1 - J(1/x) = 1/(1-x)</p><p><strong>Step 4:</strong> Solve for J(1/x):</p><p>J(1/x) = 1 - 1/(1-x) = (1-x-1)/(1-x) = -x/(1-x)</p><p><strong>Step 5:</strong> Let u = 1/x, so x = 1/u. Substitute:</p><p>J(u) = -1/u / (1 - 1/u) = (-1/u) / ((u-1)/u) = -1/(u-1) = 1/(1-u)</p><p><strong>Step 6:</strong> Replace u with x:</p><p>J(x) = 1/(1-x) = f₃(x)</p><p>∴ Answer: A</p>
Correct Answer: A