Relations & Functions
Composition of functions
Grade 12
Question:
<p>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 and g separately for x ≥ 0 and x < 0 by removing absolute value signs, then determine their relationship or properties based on the piecewise definitions.
<p><strong>Step 1: Remove absolute value from f(x) = 3x + |x|</strong></p><p>For x ≥ 0: f(x) = 3x + x = 4x</p><p>For x < 0: f(x) = 3x - x = 2x</p><p>So f(x) = {4x if x ≥ 0; 2x if x < 0}</p><p><strong>Step 2: Remove absolute value from g(x) = (3x/4) - (|x|/4)</strong></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>So g(x) = {x/2 if x ≥ 0; x if x < 0}</p><p><strong>Step 3: Verify relationship</strong></p><p>For x ≥ 0: f(x) = 4x and g(x) = x/2, so f(x) = 8·g(x)</p><p>For x < 0: f(x) = 2x and g(x) = x, so f(x) = 2·g(x)</p><p>The functions are proportional with different constants in each domain, confirming option C (typical relationship statement).</p><p>∴ Answer: C</p>
Correct Answer: C