Let f : R → R be defined as f(x) = x³ + x - 5. If g(x) is a function such that f(g(x)) = x,∀ 'x' ∈ R, then g'(63) is equal to ________________.

Solution

Maths sir · 4 steps

Do not hunt for a formula for g. Find which x makes f(x) = 63, then flip the slope there.

1. Differentiate the identity. Differentiate f(g(x)) = x with the chain rule. This gives f'(g(x)) g'(x) = 1, so g is the inverse of f.

   g'(63) = 1/f'(g(63))

2. Find g(63). Let g(63) = a, so f(a) = 63. Then a³ + a - 5 = 63, which means a³ + a = 68. Trying a = 4 gives 64 + 4 = 68, so g(63) = 4.
3. Slope of f there. f'(x) = 3x² + 1, so f'(4) = 3 × 16 + 1 = 49.
4. So. g'(63) = 1/49, which is option (a).

The idea to keep. For an inverse, g'(b) = 1/f'(a) where f(a) = b. Always find the matching input a first. Do not put b itself into f'.

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