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Find a formula for the inverse of the following function, if possible.
G(x)=root(5)(4x+2)

Find a formula for the inverse of the following function, if possible. $\newline$ $G(x)=\sqrt[5]{4 x+2}$

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Q. Find a formula for the inverse of the following function, if possible.

\newline

G(x)=\sqrt[5]{4 x+2}

Understand Goal: Understand the function and the goal. $\newline$ We are given the function $G(x) = \sqrt[5]{4x+2}$ , and we need to find its inverse. The inverse function will take the output of $G$ and return the input $x$ .
Set Equation Equal: Set $G(x)$ equal to $y$ for convenience. $\newline$ Let $y = G(x)$ , so we have $y = \sqrt[5]{4x+2}$ .
Swap $x$ and $y$ : Swap $x$ and $y$ to begin finding the inverse. $\newline$ To find the inverse, we switch the roles of $x$ and $y$ . This gives us $x = \sqrt[5]{4y+2}$ .
Isolate y Term: Isolate the term containing $y$ . To solve for $y$ , we need to get rid of the fifth root. We do this by raising both sides of the equation to the power of $5$ . This gives us $x^5 = (\sqrt[5]{4y+2})^5$ .
Simplify Equation: Simplify the equation. $\newline$ When we raise the fifth root to the power of $5$ , they cancel each other out, leaving us with $x^5 = 4y+2$ .
Solve for y: Solve for y. $\newline$ Now we need to isolate $y$ . We do this by subtracting $2$ from both sides and then dividing by $4$ . This gives us $(x^5 - 2)/4 = y$ .
Write Inverse Function: Write the inverse function. $\newline$ The inverse function, which we can denote as $G^{-1}(x)$ , is then $G^{-1}(x) = \frac{x^5 - 2}{4}$ .

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