Question

check your understanding
which of the following functions is the inverse of $f(x) = \sqrt3{x + 2}$?
\\( \circ\\ f^{-1}(x) = x^3 - 2 \\)
\\( \circ\\ f^{-1}(x) = x^3 + 8 \\)
\\( \circ\\ f^{-1}(x) = \sqrt3{x + 2} \\)
\\( \circ\\ f^{-1}(x) = (x + 2)^3 \\)

Explanation:

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

We start with the function \( f(x)=\sqrt[3]{x + 2} \). Replace \( f(x) \) with \( y \), so we have \( y=\sqrt[3]{x + 2} \).

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

To find the inverse, we swap the roles of \( x \) and \( y \). This gives us \( x=\sqrt[3]{y + 2} \).

Step3: Solve for \( y \)

To isolate \( y \), we first cube both sides of the equation to get rid of the cube root. Cubing both sides, we have \( x^{3}=(\sqrt[3]{y + 2})^{3} \), which simplifies to \( x^{3}=y + 2 \). Then, we subtract 2 from both sides to solve for \( y \): \( y=x^{3}-2 \). So the inverse function \( f^{-1}(x)=x^{3}-2 \).

Answer:

\( \boldsymbol{f^{-1}(x) = x^{3}-2} \) (corresponding to the first option: \( f^{-1}(x)=x^{3}-2 \))