Sets, Relations & Functions
Composition of functions and domain
Grade 11
Question:
<p>If \(f\) is a function with domain \([-3, 5]\) and \(g(x) = |3x + 4|\), then the domain of \((f \circ g)(x)\) is:</p>
<p>(a) \(\left(-3, \dfrac{1}{3}\right)\)</p>
<p>(b) \(\left[-3, \dfrac{1}{3}\right)\)</p>
<p>(c) \(\left[-3, \dfrac{1}{3}\right]\)</p>
<p>(d) \(\left[-3, \dfrac{-1}{3}\right]\)</p>
Step-by-Step Solution
Key Concept: For (f ∘ g)(x) to be defined, g(x) must lie within f's domain [-3, 5]. So we need -3 ≤ |3x + 4| ≤ 5, and since absolute values are non-negative, this becomes 0 ≤ |3x + 4| ≤ 5.
<p><strong>Step 1:</strong> For (f ∘ g)(x) to be defined, g(x) must be in the domain of f.</p><p><strong>Step 2:</strong> We need: -3 ≤ |3x + 4| ≤ 5</p><p><strong>Step 3:</strong> Since |3x + 4| ≥ 0 for all x, the constraint -3 ≤ |3x + 4| is always satisfied.</p><p><strong>Step 4:</strong> Solve |3x + 4| ≤ 5:</p><p>-5 ≤ 3x + 4 ≤ 5</p><p>-9 ≤ 3x ≤ 1</p><p>-3 ≤ x ≤ 1/3</p><p><strong>Step 5:</strong> The domain of (f ∘ g)(x) is [-3, 1/3]</p><p>∴ Answer: C</p>
Correct Answer: C