Relations & Functions
Invertible Functions
Grade 12
Question:
<p>Given: <i>f</i> : <i>A</i> → <i>B</i> be a function defined as \( f(x) = \dfrac{x-1}{x-2} \), where \( A = R - \{2\} \) and \( B = R - \{1\} \). So, <i>f(x)</i> is bijective function or invertible function. Then \( f^{-1}(x) \) equals:</p>
<p>\( \dfrac{2x+1}{x+1} \)</p>
<p>\( \dfrac{2x-1}{x-1} \)</p>
<p>\( \dfrac{x+1}{x-1} \)</p>
<p>\( \dfrac{2x+1}{x-1} \)</p>
Step-by-Step Solution
Key Concept: To find the inverse function, swap x and y in the equation y = f(x), then solve for y. The domain of f⁻¹ equals the range of f (which is B), and the range of f⁻¹ equals the domain of f (which is A).
<p><strong>Step 1:</strong> Start with y = f(x) = (x-1)/(x-2)</p><p><strong>Step 2:</strong> Swap x and y to get: x = (y-1)/(y-2)</p><p><strong>Step 3:</strong> Solve for y:</p><p>x(y-2) = y-1</p><p>xy - 2x = y - 1</p><p>xy - y = 2x - 1</p><p>y(x-1) = 2x - 1</p><p>y = (2x-1)/(x-1)</p><p><strong>Step 4:</strong> Verify domain: f⁻¹ has domain B = ℝ - {1} (denominator x-1 ≠ 0) ✓</p><p><strong>Step 5:</strong> Verify range: As x → 1, y → ∞; as x → ∞, y → 2. Range is ℝ - {2} = A ✓</p><p>∴ f⁻¹(x) = <strong>(2x-1)/(x-1)</strong></p>
Correct Answer: B