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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 g(x)=x38+16g(x)=\frac{x^{3}}{8}+16 ?\newlineA) g−1(x)=x−1632g^{-1}(x)=\frac{\sqrt[3]{x-16}}{2}\newlineB) g−1(x)=2x3−16g^{-1}(x)=2\sqrt[3]{x}-16\newlineC) g−1(x)=2x+163g^{-1}(x)=2\sqrt[3]{x+16}]\newlineD) g−1(x)=2x−163g^{-1}(x)=2\sqrt[3]{x-16}

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Full solution

Q. What is the inverse of the function g(x)=x38+16g(x)=\frac{x^{3}}{8}+16 ?\newlineA) g−1(x)=x−1632g^{-1}(x)=\frac{\sqrt[3]{x-16}}{2}\newlineB) g−1(x)=2x3−16g^{-1}(x)=2\sqrt[3]{x}-16\newlineC) g−1(x)=2x+163g^{-1}(x)=2\sqrt[3]{x+16}]\newlineD) g−1(x)=2x−163g^{-1}(x)=2\sqrt[3]{x-16}
  1. Swap x and y: Swap x and y to find the inverse: x=y38+16x = \frac{y^{3}}{8} + 16.
  2. Subtract 1616: Subtract 1616 from both sides: x−16=y38x - 16 = \frac{y^{3}}{8}.
  3. Multiply by 88: Multiply both sides by 88: 8(x−16)=y38(x - 16) = y^{3}.
  4. Take cube root: Take the cube root of both sides: y=8(x−16)3y = \sqrt[3]{8(x - 16)}.
  5. Simplify cube root: Simplify the cube root: y=2×x−163y = 2 \times \sqrt[3]{x - 16}.
  6. Write inverse function: Write the inverse function: g−1(x)=2⋅x−163.g^{-1}(x) = 2 \cdot \sqrt[3]{x - 16}.

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