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

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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.

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