If f(x)=(4 x+3)/(6 x-4), x ≠ 2/3 and (f ° f)(x)=g(x), where g: ℝ-{2/3} → ℝ-{2/3}, then (g ogog)(4) is equal to

Solution

Maths sir · 3 steps

Before grinding through three layers, check f(f(x)). This f undoes itself, so g is just x.

1. Build f of f. Put f(x) = (4x+3)/(6x-4) into itself. Multiply the top and the bottom by (6x-4) to clear the inner fractions.

   f(f(x)) = (4(4x+3) + 3(6x-4))/(6(4x+3) - 4(6x-4)) = 34x/34 = x

2. Read off g. So g(x) = x for every x ≠ 2/3. This works because f has the form (ax+b)/(cx-a), and every such function is its own inverse.
3. So. Applying the identity three times changes nothing: g(g(g(4))) = 4. The answer is (d).

The idea to keep. For f(x) = (ax+b)/(cx+d) with a + d = 0, f is its own inverse and f(f(x)) = x. Spot this first and repeated compositions become trivial.

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