Question

find the inverse of the function.
h(x) = \sqrt3{x} - 2
write your answer in the form a(bx + c)^3 + d, where a, b, c, and d are constants. simplify any fractions.
h^{-1}(x) = \boxed{}

Explanation:

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

We start by writing the function as \( y = \sqrt[3]{x} - 2 \).

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

To find the inverse, we interchange \( x \) and \( y \), so we get \( x = \sqrt[3]{y} - 2 \).

Step3: Solve for \( y \)

First, add 2 to both sides of the equation: \( x + 2 = \sqrt[3]{y} \). Then, cube both sides to eliminate the cube root: \( (x + 2)^3 = y \).

Step4: Rewrite in the required form

The equation \( y=(x + 2)^3\) can be written as \( y = 1(1x + 2)^3+0 \) which is in the form \( a(bx + c)^3 + d \) where \( a = 1 \), \( b = 1 \), \( c = 2 \) and \( d = 0 \). So the inverse function \( h^{-1}(x)=(x + 2)^3 \)

Answer:

\( (x + 2)^3 \)