Question

find the inverse of the one-to-one function. f(x) = (x - 4)^3 f^{-1}(x) = \square (type an exact answer, using radicals as needed.)

Explanation:

Step1: Replace f(x) with y

We start by writing the function as \( y=(x - 4)^3 \).

Step2: Swap x and y

Interchange the roles of x and y to get \( x=(y - 4)^3 \).

Step3: Solve for y

Take the cube root of both sides: \( \sqrt[3]{x}=y - 4 \). Then, add 4 to both sides to isolate y: \( y=\sqrt[3]{x}+4 \).

Step4: Replace y with \( f^{-1}(x) \)

We get \( f^{-1}(x)=\sqrt[3]{x}+4 \).

Answer:

\( \sqrt[3]{x}+4 \)