Find the Inverse y=6-x^3

$y = 6 - x^{3}$

Interchange the variables.

$x = 6 - y^{3}$

Solve for $y$ .

Rewrite the equation as $6 - y^{3} = x$ .

$6 - y^{3} = x$

Subtract $6$ from both sides of the equation.

$- y^{3} = x - 6$

Divide each term in $- y^{3} = x - 6$ by $- 1$ and simplify.

Divide each term in $- y^{3} = x - 6$ by $- 1$ .

$\frac{- y^{3}}{- 1} = \frac{x}{- 1} + \frac{- 6}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{y^{3}}{1} = \frac{x}{- 1} + \frac{- 6}{- 1}$

Divide $y^{3}$ by $1$ .

$y^{3} = \frac{x}{- 1} + \frac{- 6}{- 1}$

Simplify the right side.

Simplify each term.

Move the negative one from the denominator of $\frac{x}{- 1}$ .

$y^{3} = - 1 \cdot x + \frac{- 6}{- 1}$

Rewrite $- 1 \cdot x$ as $- x$ .

$y^{3} = - x + \frac{- 6}{- 1}$

Divide $- 6$ by $- 1$ .

$y^{3} = - x + 6$

Take the cube root of both sides of the equation to eliminate the exponent on the left side.

$y = \sqrt[3]{- x + 6}$

Replace $y$ with $f^{- 1} (x)$ to show the final answer.

$f^{- 1} (x) = \sqrt[3]{- x + 6}$

Verify if $f^{- 1} (x) = \sqrt[3]{- x + 6}$ is the inverse of $f (x) = 6 - x^{3}$ .

To verify the inverse, check if $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ .

Evaluate $f^{- 1} (f (x))$ .

Set up the composite result function.

$f^{- 1} (f (x))$

Evaluate $f^{- 1} (6 - x^{3})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (6 - x^{3}) = \sqrt[3]{- (6 - x^{3}) + 6}$

Apply the distributive property.

$f^{- 1} (6 - x^{3}) = \sqrt[3]{- 1 \cdot 6 + x^{3} + 6}$

Multiply $- 1$ by $6$ .

$f^{- 1} (6 - x^{3}) = \sqrt[3]{- 6 + x^{3} + 6}$

Add $- 6$ and $6$ .

$f^{- 1} (6 - x^{3}) = \sqrt[3]{x^{3} + 0}$

Add $x^{3}$ and $0$ .

$f^{- 1} (6 - x^{3}) = \sqrt[3]{x^{3}}$

Pull terms out from under the radical, assuming real numbers.

$f^{- 1} (6 - x^{3}) = x$

Evaluate $f (f^{- 1} (x))$ .

Set up the composite result function.

$f (f^{- 1} (x))$

Evaluate $f (\sqrt[3]{- x + 6})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\sqrt[3]{- x + 6}) = 6 - {(\sqrt[3]{- x + 6})}^{3}$

Simplify each term.

Rewrite ${\sqrt[3]{- x + 6}}^{3}$ as $- x + 6$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{- x + 6}$ as ${(- x + 6)}^{\frac{1}{3}}$ .

$f (\sqrt[3]{- x + 6}) = 6 - {({(- x + 6)}^{\frac{1}{3}})}^{3}$

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$f (\sqrt[3]{- x + 6}) = 6 - {(- x + 6)}^{\frac{1}{3} \cdot 3}$

Combine $\frac{1}{3}$ and $3$ .

$f (\sqrt[3]{- x + 6}) = 6 - {(- x + 6)}^{\frac{3}{3}}$

Cancel the common factor of $3$ .

Cancel the common factor.

$f (\sqrt[3]{- x + 6}) = 6 - {(- x + 6)}^{\frac{3}{3}}$

Rewrite the expression.

$f (\sqrt[3]{- x + 6}) = 6 - (- x + 6)$

Simplify.

$f (\sqrt[3]{- x + 6}) = 6 - (- x + 6)$

Apply the distributive property.

$f (\sqrt[3]{- x + 6}) = 6 + x - 1 \cdot 6$

Multiply $- - x$ .

Multiply $- 1$ by $- 1$ .

$f (\sqrt[3]{- x + 6}) = 6 + 1 x - 1 \cdot 6$

Multiply $x$ by $1$ .

$f (\sqrt[3]{- x + 6}) = 6 + x - 1 \cdot 6$

Multiply $- 1$ by $6$ .

$f (\sqrt[3]{- x + 6}) = 6 + x - 6$

Combine the opposite terms in $6 + x - 6$ .

Subtract $6$ from $6$ .

$f (\sqrt[3]{- x + 6}) = x + 0$

Add $x$ and $0$ .

$f (\sqrt[3]{- x + 6}) = x$

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \sqrt[3]{- x + 6}$ is the inverse of $f (x) = 6 - x^{3}$ .

$f^{- 1} (x) = \sqrt[3]{- x + 6}$

Find the Inverse y=6-x^3

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