Relations & Functions
Composite functions
Grade 12
Question:
<p>Let \(f(x)\) and \(g(x)\) are two functions defined from \(R^+ \to R\) such that</p><p>\[f(x) = \begin{cases} 1 - \sqrt{x}, & \text{if } x \text{ is rational} \\ x^2, & \text{if } x \text{ is irrational} \end{cases}\] and \[g(x) = \begin{cases} x, & \text{if } x \text{ is rational} \\ 1 - x, & \text{if } x \text{ is irrational} \end{cases}\]</p><p>The composite function \(f(g(x))\) is:</p>
<p>(a) one-one onto</p>
<p>(b) many one into</p>
<p>(c) one-one into</p>
<p>(d) many one onto</p>
Step-by-Step Solution
Key Concept: To find f(g(x)), you must first determine whether g(x) is rational or irrational based on whether x is rational or irrational, then apply the appropriate case of f to that result.
<p><strong>Step 1:</strong> Analyze g(x) for rational and irrational x.</p><p>• When x is rational: g(x) = x (rational)</p><p>• When x is irrational: g(x) = 1 - x (irrational, since 1 minus an irrational is irrational)</p><p><strong>Step 2:</strong> Apply f to g(x) for each case.</p><p><strong>Case 1 - x is rational:</strong> g(x) = x is rational, so f(g(x)) = f(x) = 1 - √x</p><p><strong>Case 2 - x is irrational:</strong> g(x) = 1 - x is irrational, so f(g(x)) = [g(x)]² = (1 - x)²</p><p><strong>Step 3:</strong> Write the composite function.</p><p>$$f(g(x)) = \begin{cases} 1 - \sqrt{x}, & \text{if } x \text{ is rational} \\ (1-x)^2, & \text{if } x \text{ is irrational} \end{cases}$$</p><p>∴ Answer: D</p>
Correct Answer: D