Answer
The correct match that best describes the inverse of the equation $y=5{{x}^{3}}+10$ is option (C) $y=\sqrt[3]{\frac{x-10}{5}}$.
The correct match that best describes the inverse of the equation $y={{\left( 5x+10 \right)}^{3}}$ is option (A) $y=\frac{\sqrt[3]{x}-10}{5}$.
The correct match that best describes the inverse of the equation $y=5{{\left( x+10 \right)}^{3}}$ is option (B)$y=\sqrt[3]{\frac{x}{5}}-10$.
The correct match that best describes the inverse of the equation $y={{\left( 5x \right)}^{3}}+10$ is option (D) $y=\frac{\sqrt[3]{x-10}}{5}$.
Work Step by Step
Consider the first provided function,
$y=5{{x}^{3}}+10$
Now, for the inverse of the above function, first show the provided function is one-one
Subtract $10$ from both sides of the above equation
$y-10=5{{x}^{3}}$
Divide both side by $5$
$\frac{y-10}{5}={{x}^{3}}$
Take the cube root on both sides of the above equation
$\sqrt[3]{\frac{y-10}{5}}=x$
Switching x and y:
$\sqrt[3]{\frac{x-10}{5}}=y$
Or
$y=\sqrt[3]{\frac{x-10}{5}}$
Thus, the correct option is (C). $y=\sqrt[3]{\frac{x-10}{5}}$.
Consider the second provided function,
$y={{\left( 5x+10 \right)}^{3}}$
Take the cube root on both side of the above equation
$\sqrt[3]{y}=5x+10$
Subtract $10$ from both sides of the above equation
$\sqrt[3]{y}-10=5x$
Divide both sides by $5$
$\frac{\sqrt[3]{y}-10}{5}=x$
Switch x and y:
$\frac{\sqrt[3]{x}-10}{5}=y$
Or
$y=\frac{\sqrt[3]{x}-10}{5}$
Therefore, the correct option is (A). $y=\frac{\sqrt[3]{x}-10}{5}$.
Consider the third provided function:
$y=5{{\left( x+10 \right)}^{3}}$
Divide both side by $5$,
$\frac{y}{5}={{\left( x+10 \right)}^{3}}$
Take the cube root on both side of the above equation,
$\sqrt[3]{\frac{y}{5}}=x+10$
Subtract $10$ from both sides of the above equation,
$\sqrt[3]{\frac{y}{5}}-10=x$
Switch x and y:
$\sqrt[3]{\frac{x}{5}}-10=y$
Or
$y=\sqrt[3]{\frac{x}{5}}-10$
Thus, the correct option is (B). $y=\sqrt[3]{\frac{x}{5}}-10$.
Consider the fourth provided function:
$y={{\left( 5x \right)}^{3}}+10$
Subtract $10$ from both sides of the above equation
$y-10={{\left( 5x \right)}^{3}}$
Take the cube root on both sides of the above equation
$\sqrt[3]{y-10}=5x$
Divide both sides by $5$
$\frac{\sqrt[3]{y-10}}{5}=x$
Switch x and y:
$\frac{\sqrt[3]{x-10}}{5}=y$
Or
$y=\frac{\sqrt[3]{x-10}}{5}$
Thus, the correct option is (D). $y=\frac{\sqrt[3]{x-10}}{5}$.
