Relations & Functions
Piecewise Functions
Grade 12
Question:
<p>\[f(x) = \begin{cases} x, & \text{if } x \text{ is rational} \\ 0, & \text{if } x \text{ is irrational} \end{cases}, \quad g(x) = \begin{cases} 0, & \text{if } x \text{ is rational} \\ x, & \text{if } x \text{ is irrational} \end{cases}\]\nThen, \(f \circ g\) is</p>
<p>(a) one-one and into</p>
<p>(b) neither one-one nor onto</p>
<p>(c) many one and onto</p>
<p>(d) one-one and onto</p>
Step-by-Step Solution
Key Concept: To find (f ∘ g)(x) = f(g(x)), we must first evaluate g(x), then apply f to that result. The key is recognizing that g(x) produces either 0 (rational) or x (irrational), and then determining what f does to each output.
<p><strong>Step 1: Find the composition f ∘ g</strong></p><p>We need to find (f ∘ g)(x) = f(g(x)) for all x ∈ ℝ.</p><p><strong>Step 2: Evaluate g(x) for rational x</strong></p><p>If x is rational: g(x) = 0 (which is rational)</p><p>Then f(g(x)) = f(0) = 0 (since 0 is rational)</p><p><strong>Step 3: Evaluate g(x) for irrational x</strong></p><p>If x is irrational: g(x) = x (which is irrational)</p><p>Then f(g(x)) = f(x) = 0 (since x is irrational)</p><p><strong>Step 4: Express (f ∘ g)(x)</strong></p><p>(f ∘ g)(x) = f(g(x)) = 0 for all x ∈ ℝ</p><p><strong>Step 5: Check if f ∘ g is one-one</strong></p><p>Since (f ∘ g)(x) = 0 for all x, every element in the domain maps to 0. For example, f(∘g)(1) = 0 and (f ∘ g)(√2) = 0. Multiple inputs give the same output, so f ∘ g is NOT one-one (it is many-one).</p><p><strong>Step 6: Check if f ∘ g is onto</strong></p><p>The range of f ∘ g is {0}, but the codomain is ℝ. Since only 0 is in the range and ℝ contains other elements like 1, 2, π, etc., f ∘ g is NOT onto (it is into).</p><p><strong>∴ Answer:</strong> b</p>
Correct Answer: b