Question

find the inverse of the function.\\( f(x) = x^3 + 6 \\)\\( \text{write your answer in the form } a \cdot \sqrt3{bx + c} + d, \text{ where } a, b, c, \text{ and } d \text{ are constants. simplify any fractions.} \\)\\( f^{-1}(x) = \\)

Explanation:

Step1: Let \( y = f(x) = x^3 + 6 \)

We start by setting \( y \) equal to the function \( f(x) \). This is the first step in finding the inverse function, as we need to express \( x \) in terms of \( y \).

Step2: Solve for \( x \) in terms of \( y \)

Subtract 6 from both sides: \( y - 6 = x^3 \)
Then take the cube root of both sides: \( x=\sqrt[3]{y - 6} \)

Step3: Replace \( x \) with \( f^{-1}(x) \) and \( y \) with \( x \)

This gives us the inverse function: \( f^{-1}(x)=\sqrt[3]{x - 6} \)
In the form \( a\cdot\sqrt[3]{bx + c}+d \), we have \( a = 1 \), \( b = 1 \), \( c=- 6 \), and \( d = 0 \), so \( f^{-1}(x)=1\cdot\sqrt[3]{1\cdot x+(-6)}+0=\sqrt[3]{x - 6} \)

Answer:

\( \sqrt[3]{x - 6} \)