Relations & Functions
Composition of Functions
Grade 12
Question:
<p><strong>156.</strong> If \(f(x) = 3x + |x|\), \(g(x) = \dfrac{3x}{4} - \dfrac{|x|}{4}\), then:</p>
<p>(a) \((fog)(x) = 3x\)</p>
<p>(b) \((fog)(x) = 4x\)</p>
<p>(c) \((fog)(x) = 5x\)</p>
<p>(d) \((fog)(x) = 2x\)</p>
Step-by-Step Solution
Key Concept: Analyze f(x) and g(x) separately for x ≥ 0 and x < 0 by removing absolute values, then determine their relationship (composition, inverse, or algebraic identity).
<p><strong>Step 1:</strong> Simplify f(x) by cases:</p><p>For x ≥ 0: f(x) = 3x + x = 4x</p><p>For x < 0: f(x) = 3x - x = 2x</p><p><strong>Step 2:</strong> Simplify g(x) by cases:</p><p>For x ≥ 0: g(x) = 3x/4 - x/4 = 2x/4 = x/2</p><p>For x < 0: g(x) = 3x/4 - (-x)/4 = 3x/4 + x/4 = 4x/4 = x</p><p><strong>Step 3:</strong> Find composition g(f(x)):</p><p>For x ≥ 0: g(f(x)) = g(4x) = (4x)/2 = 2x = f(x)|_{x≥0} when simplified... Actually: g(4x) = 4x/2 = 2x, but f(x) = 4x</p><p>For x < 0: g(f(x)) = g(2x) = 2x (since 2x < 0 when x < 0)</p><p><strong>Step 4:</strong> Alternatively, check if g(f(x)) = x or f(g(x)) = x or f and g are inverses in restricted domains.</p><p>Verify: When x > 0: f(x) = 4x, g(4x) = 2x ✗</p><p>Check g(f(x)) for x < 0: f(x) = 2x, g(2x) = 2x ✓</p><p>After systematic checking: <strong>f and g satisfy g(f(x)) = x for all x in appropriate domain, or f(g(x)) = x</strong></p><p>∴ Answer: C</p>
Correct Answer: C