Let f, g:(1, ∞) → ℝ be defined as f(x)=(2 x+3)/(5 x+2) and g(x)=(2-3 x)/(1-x). If the range of the function fog: [2,4] → ℝ is [α, β], then 1/(β-α) is equal to

Solution

Maths sir · 5 steps

Simplify fog to one fraction, check it is increasing, and the range comes from the two endpoints.

1. Rewrite g. Multiply the top and bottom of g(x) by -1 to get g(x) = (3x-2)/(x-1). On [2,4], g takes values between 10/3 and 4. These lie inside (1, ∞), so f(g(x)) is defined there.
2. Build fog. 2g+3 = (2(3x-2)+3(x-1))/(x-1) = (9x-7)/(x-1) and 5g+2 = (5(3x-2)+2(x-1))/(x-1) = (17x-12)/(x-1). Divide them and the x-1 cancels.

   (f° g)(x) = (9x-7)/(17x-12)

3. Check monotonic. For (ax+b)/(cx+d), the derivative is (ad-bc)/(cx+d)². Here ad-bc = 9(-12)-(-7)(17) = -108+119 = 11 > 0, so fog is increasing on [2,4]. The smallest value is at x=2 and the largest is at x=4.
4. Endpoint values. α = (18-7)/(34-12) = 11/22 = 1/2 and β = (36-7)/(68-12) = 29/56. So β-α = 29/56-28/56 = 1/56.
5. So. 1/(β-α) = 56, which is option (a).

The idea to keep. Composing two linear-over-linear functions gives another one, and that is always monotonic on an interval where it is defined. So its range on [p,q] is fixed by f(p) and f(q).

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