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$What is the inverse of the function g(x)=(x^(3))/(8)+16 ? {:[g^(-1)(x)=(root(3)(x-16))/(2)],[g^(-1)(x)=2root(3)(x)-16],[g^(-1)(x)=2root(3)(x+16)],[g^(-1)(x)=2root(3)(x-16)]:}$

What is the inverse of the function $\newline$ $g(x)=\frac{x^{3}}{8}+16$ ? $\newline$ $g^{-1}(x)=\frac{\sqrt[3]{x-16}}{2}$ $\newline$ $g^{-1}(x)=2\sqrt[3]{x}-16$ $\newline$ $g^{-1}(x)=2\sqrt[3]{x+16}$ $\newline$ $g^{-1}(x)=2\sqrt[3]{x-16}$

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Find the vertex of the transformed function

Full solution

Q. What is the inverse of the function

\newline

g(x)=\frac{x^{3}}{8}+16

?

\newline

g^{-1}(x)=\frac{\sqrt[3]{x-16}}{2}

\newline

g^{-1}(x)=2\sqrt[3]{x}-16

\newline

g^{-1}(x)=2\sqrt[3]{x+16}

\newline

g^{-1}(x)=2\sqrt[3]{x-16}

Swap x and y: Swap x and y to find the inverse: $x = \frac{y^{3}}{8} + 16$ .
Subtract $16$ : Subtract $16$ from both sides: $x - 16 = \frac{y^{3}}{8}$ .
Multiply by $8$ : Multiply both sides by $8$ : $8(x - 16) = y^{3}$ .
Take cube root: Take the cube root of both sides: $y = \sqrt[3]{8(x - 16)}$ .
Simplify cube root: Simplify the cube root: $y = 2 \times \sqrt[3]{x - 16}$ .
Write inverse function: Write the inverse function: $g^{-1}(x) = 2 \cdot \sqrt[3]{x - 16}.$

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