Let f be a function such that f(x)+3 f(24/x)=4 x, x ≠ 0. Then f(3)+f(8) is equal to

Solution

Maths sir · 4 steps

Note that 3 and 8 swap under x → 24/x. Write both equations and just add them.

1. Spot the pair. Since 3 × 8 = 24, putting x = 3 gives 24/x = 8, and putting x = 8 gives 24/x = 3. So the two values we need appear together in both equations.
2. Two equations. With x = 3: f(3) + 3f(8) = 12. With x = 8: f(8) + 3f(3) = 32.

   4[f(3) + f(8)] = 12 + 32 = 44

3. Quick check. Solving the pair separately gives f(8) = 1/2 and f(3) = 21/2. Then f(3) + 3f(8) = 12 and f(8) + 3f(3) = 32, as required.
4. So. f(3) + f(8) = 44/4 = 11, which is option (b).

The idea to keep. When a functional equation links f(x) with f(k/x), substitute both partners of the pair. You get two equations in two unknowns, and adding them often gives the sum at once.

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