Sets, Relations & Functions
Composite Functions
Grade 11
Question:
<p>Given \(f(x) = x^2,\ x \in \mathbb{R}\), \(g(A) = \{x \in \mathbb{R} : f(x) \in A\}\), \(S \equiv [0, 4]\). If \(g(s) = \{x \in \mathbb{R} : 0 \le x^2 \le 4\}\), which of the following is correct?</p>
<p>\(g(f(s)) = g(s)\) is correct</p>
<p>\(f(g(s)) = f(s)\) is correct</p>
<p>\(g(f(s)) = g(s)\) is incorrect</p>
<p>\(f(g(s)) = f(s)\) is incorrect</p>
Step-by-Step Solution
Key Concept: The function g(A) represents the preimage (inverse image) of set A under f, meaning g(A) contains all x values whose images lie in A. For f(x) = x², finding g([0,4]) requires solving 0 ≤ x² ≤ 4, which gives both positive AND negative x values.
<p><strong>Step 1:</strong> Understand the definition. g(A) = {x ∈ ℝ : f(x) ∈ A} is the preimage (inverse image) of set A under function f.</p><p><strong>Step 2:</strong> Given f(x) = x² and S = [0, 4], we need g(S) = {x ∈ ℝ : f(x) ∈ [0, 4]}.</p><p><strong>Step 3:</strong> This means we solve: 0 ≤ x² ≤ 4</p><p><strong>Step 4:</strong> From x² ≥ 0, this is always true for all real x. From x² ≤ 4, we get |x| ≤ 2, which means -2 ≤ x ≤ 2.</p><p><strong>Step 5:</strong> Therefore, g(S) = [-2, 2], which is the interval containing all x values whose squares fall in [0, 4].</p><p>∴ Answer: C</p>
Correct Answer: C