Let f(x) = (x − 2)^3 + 4. Is f one-to-one?. Determine f^-1 (x). Is f ^−1 (x) continuous? Determine x such that f^−1 is differentiable at x and evaluate its derivative, (f ^−1 )'(x)

Question

must be solved without calculator. Thanks!

Answer 1

f is strictly increasing on the whole real line; therefore, f is one-to-one.

To find f^-1(x) interchange the roles of x and y, i.e.

x = (y-2)³ + 4 and then y = 2 +(x-4)^1/3

f^-1(x) is continuous on the whole real line

and the derivative of f^-1(x) is (1/3)(x-4)^-2/3