Let f(x) and g(x) be two real polynomials of degree 2 and 1 respectively. If f(g(x)) = 8x² - 2x and g(f(x)) = 4x² + 6x + 1, then the value of f(2) + g(2) is _________.

Solution

Maths sir · 5 steps

Write f and g with unknown coefficients, then match coefficients in both composites, the easy one first.

1. Set up. Take f(x) = ax² + bx + c with a ≠ 0 and g(x) = px + q with p ≠ 0. Five unknowns, and each composite gives three coefficient equations.
2. Use g(f(x)) first. g(f(x)) = pa x² + pb x + (pc + q). Matching with 4x² + 6x + 1 gives ap = 4, bp = 6 and pc + q = 1.

   f(g(x)) = ap²x² + (2apq + bp) x + (aq² + bq + c) = 8x² - 2x

3. Match coefficients. From x²: ap² = 8. Divide by ap = 4 to get p = 2, so a = 2 and b = 3. From x: 8q + 6 = -2, so q = -1. From the constant: 2 - 3 + c = 0, so c = 1.
4. Check. The last equation from g(f(x)) is pc + q = 2 - 1 = 1, which matches. So f(x) = 2x² + 3x + 1 and g(x) = 2x - 1.
5. So. f(2) = 8 + 6 + 1 = 15 and g(2) = 4 - 1 = 3. Their sum is 18.

The common slip: 15. Stopping after finding f(2) = 15 and forgetting to add g(2) = 3 gives 15, not 18.

The idea to keep. When the degrees are given, write the polynomials with unknown coefficients and compare coefficients. Dividing two leading-coefficient equations, like ap² by ap, often unlocks everything.

Answer from the NTA final answer key. Solution by PrepWiser.