Questions: f(x) = ∛x - 2 f^(-1)(x) = □ (Type your answer in factored form.)

Solution Steps

To find the inverse of the function \( f(x) = \sqrt[3]{x} - 2 \), we need to follow these steps:

1. Replace \( f(x) \) with \( y \).
2. Swap \( x \) and \( y \) to solve for \( y \) in terms of \( x \).
3. Isolate \( y \) to find the inverse function \( f^{-1}(x) \).

We start with the function given by \( f(x) = \sqrt[3]{x} - 2 \).

Let \( y = f(x) \), which gives us the equation: \[ y = \sqrt[3]{x} - 2 \] Next, we swap \( x \) and \( y \): \[ x = \sqrt[3]{y} - 2 \]

To isolate \( y \), we first add 2 to both sides: \[ x + 2 = \sqrt[3]{y} \] Next, we cube both sides to eliminate the cube root: \[ (y) = (x + 2)^3 \]

Final Answer