Algebra
Functions, Composition of Functions
jee_main_2026_jan_21_shift_1
Grade None
Question:
If g(x) = 3x² + 2x - 3, f(0) = -3 and 4g(f(x)) = 3x² - 32x + 72, then f(g(2)) is equal to:
A. 5/2
B. 7/2
C. 9/2
D. 11/2
Step-by-Step Solution
Key Concept: Solve 4g(f(x)) = 3x² - 32x + 72 to find f(x).
Step 1: 4g(f(x)) = 12f(x)² + 8f(x) - 12 = 3x² - 32x + 72. Step 2: 12f(x)² + 8f(x) - 3x² + 32x - 84 = 0. Step 3: Let f(x) = mx + n (linear, since coefficients force quadratic term to zero). Step 4: 12(mx+n)² + 8(mx+n) = 3x² - 32x + 84. Step 5: 12m²x² + 24mnx + 12n² + 8mx + 8n = 3x² - 32x + 84. Step 6: 12m² = 3 => m = ±1/2. 24mn + 8m = -32 => If m = 1/2: 12n + 4 = -32 => n = -3. Step 7: f(x) = x/2 - 3. Check f(0) = -3 ✓. Step 8: g(2) = 3(4) + 4 - 3 = 13. f(g(2)) = f(13) = 13/2 - 3 = 7/2.
Correct Answer: B
Confused by the solution? Ask the AI to explain a specific step, tell you where you went wrong, or break down the key trap in this question.