If g(x)= inverse of f(x) find g'(x)

Question

The function f(x) = 3x^3 + 2x + 1 is invertable. If g(x) = f^−1(x), find g'(1).

Answer 1

To find g'(1), we will use the Inverse Function Theorem.

Inverse Function Theorem If y = g(x) is the inverse of f(x), then g' (x) = ¹/_{f ' (g(x))}.

Since, f(x) = 3x³ + 2x + 1, it follows that f ' (x) = 9x² + 2. So by the above theorem,

g' (1) = ¹/_{f ' (g(1))} where f ' (g(1)) = 9 · g(1)² + 2.

It remains to figure out what g(1) equals. Since g(x) = f ^-1 (x), g(1) is the solution to

3x³ + 2x + 1 = 1 <----> (3x² + 2)x = 0

The only solution to this is x = 0. Therefore, g(1) = 0 and g' (1) = g' (1) = ¹/_{f ' (0)} = ¹/₂.