Question

find a formula for the inverse of the following function, if possible.
$g(x) = (4x + 2)^{\frac{1}{3}}$
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$g^{-1}(x) = $
☐ does not have an inverse function

Explanation:

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

We start by letting \( y = G(x) \), so \( y=(4x + 2)^{\frac{1}{3}} \).

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

To find the inverse, we swap the roles of \( x \) and \( y \). This gives us \( x=(4y + 2)^{\frac{1}{3}} \).

Step3: Cube both sides

To eliminate the cube - root, we cube both sides of the equation. Cubing the left - hand side gives \( x^{3} \), and cubing the right - hand side: \( ((4y + 2)^{\frac{1}{3}})^{3}=4y + 2 \). So we have the equation \( x^{3}=4y + 2 \).

Step4: Solve for \( y \)

First, subtract 2 from both sides of the equation: \( x^{3}-2 = 4y \). Then, divide both sides by 4 to solve for \( y \): \( y=\frac{x^{3}-2}{4} \).

Step5: Replace \( y \) with \( G^{-1}(x) \)

Since \( y \) represents the inverse function of \( G(x) \), we write \( G^{-1}(x)=\frac{x^{3}-2}{4} \).

Answer:

\( G^{-1}(x)=\frac{x^{3}-2}{4} \)