Let f(x) = x^2, x \in \mathbb{R}. For any A \subseteq \mathbb{R}, define g(A) = \{x \in \mathbb{R} : f(x) \in A\}. If S = [0, 4], then which one of the following statements is not true ?
Step-by-Step Solution
Key Concept: Understand that g(A) gives all x-values whose squares lie in A, while f maps x-values to their squares. The inverse relationship between f and g determines which statements are true.
<p><strong>Step 1: Understand the functions</strong></p><p>Given: f(x) = x², x ∈ ℝ and S = [0, 4]</p><p>g(A) = {x ∈ ℝ : f(x) ∈ A} = {x ∈ ℝ : x² ∈ A}</p><p></p><p><strong>Step 2: Find g(S)</strong></p><p>g(S) = {x ∈ ℝ : x² ∈ [0, 4]}</p><p>This means 0 ≤ x² ≤ 4, so -2 ≤ x ≤ 2</p><p>Therefore: g(S) = [-2, 2]</p><p></p><p><strong>Step 3: Evaluate Option A: f(g(S)) ≠ f(S)</strong></p><p>f(g(S)) = f([-2, 2]) = {x² : x ∈ [-2, 2]} = [0, 4] = S</p><p>f(S) = f([0, 4]) = {x² : x ∈ [0, 4]} = [0, 16]</p><p>So f(g(S)) = [0, 4] ≠ [0, 16] = f(S) ✓ (Statement A is TRUE)</p><p></p><p><strong>Step 4: Evaluate Option B: f(g(S)) = S</strong></p><p>From Step 3: f(g(S)) = [0, 4] = S ✓ (Statement B is TRUE)</p><p></p><p><strong>Step 5: Evaluate Option C: g(f(S)) = g(S)</strong></p><p>f(S) = f([0, 4]) = [0, 16] (computed in Step 3)</p><p>g(f(S)) = g([0, 16]) = {x ∈ ℝ : x² ∈ [0, 16]} = {x : 0 ≤ x² ≤ 16} = [-4, 4]</p><p>g(S) = [-2, 2] (from Step 2)</p><p>So g(f(S)) = [-4, 4] ≠ [-2, 2] = g(S) ✗ (Statement C is FALSE)</p><p></p><p><strong>Step 6: Evaluate Option D: g(f(S)) ≠ S</strong></p><p>From Step 5: g(f(S)) = [-4, 4] ≠ [0, 4] = S ✓ (Statement D is TRUE)</p><p></p><p><strong>∴ Answer: 3</strong></p>
Correct Answer: 3