Question

find the inverse of the function.

( g(x) = (x - 6)^3 )

write your answer in the form ( a cdot sqrt3{bx + c} + d ), where ( a ), ( b ), ( c ), and ( d ) are constants. simplify any fractions.

( g^{-1}(x) = )

there are some input buttons below the answer box, including fraction, cube root, parentheses, exponent 3, plus-minus buttons, which are omitted as they are part of the input interface

Explanation:

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

We start by letting \( y = g(x) \), so \( y=(x - 6)^{3} \).

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

To find the inverse, we swap the roles of \( x \) and \( y \). This gives us \( x=(y - 6)^{3} \).

Step3: Solve for \( y \)

We take the cube - root of both sides to isolate the term with \( y \). The cube - root of \( x \) is \( \sqrt[3]{x} \), and the cube - root of \( (y - 6)^{3} \) is \( y - 6 \). So we have \( \sqrt[3]{x}=y - 6 \).
Then, we add 6 to both sides to solve for \( y \). \( y=\sqrt[3]{x}+6 \). In the form \( a\cdot\sqrt[3]{bx + c}+d \), we can see that \( a = 1 \), \( b = 1 \), \( c = 0 \), and \( d = 6 \). So \( g^{-1}(x)=\sqrt[3]{x}+6 \) or \( 1\cdot\sqrt[3]{1x + 0}+6 \).

Answer:

\( \sqrt[3]{x}+6 \) (or in the required form \( 1\cdot\sqrt[3]{1x + 0}+6 \))