For some a, b, c ∈ℕ, let f(x) = ax - 3 and g(x)=xᵇ+c,x∈ℝ. If (fog)⁻¹(x) = ( (x - 7)/2 )^(1/3), then (fog)(ac) + (gof)(b) is equal to ____________.

Solution

Maths sir · 5 steps

Invert the given inverse first: that hands you f ° g directly, and then you can read off a, b, c.

1. Undo the inverse. Put y = ((x-7)/2)^(1/3). Cubing gives y³ = (x-7)/2, so x = 2y³ + 7. Therefore (f ° g)(x) = 2x³ + 7.

   (f ° g)(x) = a(xᵇ + c) - 3 = a xᵇ + (ac - 3) = 2x³ + 7

2. Match coefficients. Comparing both sides gives a = 2, b = 3 and ac - 3 = 7. So ac = 10 and c = 5, all natural numbers as required.
3. First term. (f ° g)(ac) = (f ° g)(10) = 2 × 10³ + 7 = 2007.
4. Second term. (g ° f)(3) = g(f(3)). First f(3) = 2 × 3 - 3 = 3, then g(3) = 3³ + 5 = 32.
5. So. (f ° g)(ac) + (g ° f)(b) = 2007 + 32 = 2039.

The common slip: 2068. Working out (g ° f)(3) as f(g(3)) = 61 instead of g(f(3)) = 32 gives 2068, not 2039.

The idea to keep. When a question gives you an inverse, find the function itself first, then compare coefficients. Remember that g ° f means apply f first, then g.

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