Question

find the inverse function.
$f(x) = x^3 - 12$
$f^{-1}(x) = ?\sqrt{x + \square}$

Explanation:

Step1: Replace \( f(x) \) with \( y \)

We start by writing the function as \( y = x^3 - 12 \).

Step2: Swap \( x \) and \( y \)

To find the inverse, we swap the roles of \( x \) and \( y \), so we get \( x = y^3 - 12 \).

Step3: Solve for \( y \)

First, add 12 to both sides of the equation: \( x + 12 = y^3 \). Then, take the cube root of both sides to solve for \( y \). The cube root of \( y^3 \) is \( y \), and the cube root of \( x + 12 \) is \( \sqrt[3]{x + 12} \). So, \( y=\sqrt[3]{x + 12} \).

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

We now write \( f^{-1}(x)=\sqrt[3]{x + 12} \).

Answer:

\( f^{-1}(x)=\sqrt[3]{x + 12} \) (So the number in the "?" is 3 and the number in the box is 12)