Algebra Examples

$2 + \sqrt[3]{x - 1}$

Step 1

Interchange the variables.

$x = 2 + \sqrt[3]{y - 1}$

Step 2

Solve for $y$ .

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Step 2.1

Rewrite the equation as $2 + \sqrt[3]{y - 1} = x$ .

$2 + \sqrt[3]{y - 1} = x$

Step 2.2

Subtract $2$ from both sides of the equation.

$\sqrt[3]{y - 1} = x - 2$

Step 2.3

To remove the radical on the left side of the equation, cube both sides of the equation.

${\sqrt[3]{y - 1}}^{3} = {(x - 2)}^{3}$

Step 2.4

Simplify each side of the equation.

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Step 2.4.1

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

${({(y - 1)}^{\frac{1}{3}})}^{3} = {(x - 2)}^{3}$

Step 2.4.2

Simplify the left side.

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Step 2.4.2.1

Simplify ${({(y - 1)}^{\frac{1}{3}})}^{3}$ .

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Step 2.4.2.1.1

Multiply the exponents in ${({(y - 1)}^{\frac{1}{3}})}^{3}$ .

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Step 2.4.2.1.1.1

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

${(y - 1)}^{\frac{1}{3} \cdot 3} = {(x - 2)}^{3}$

Step 2.4.2.1.1.2

Cancel the common factor of $3$ .

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Step 2.4.2.1.1.2.1

Cancel the common factor.

${(y - 1)}^{\frac{1}{3} \cdot 3} = {(x - 2)}^{3}$

Step 2.4.2.1.1.2.2

Rewrite the expression.

${(y - 1)}^{1} = {(x - 2)}^{3}$

Step 2.4.2.1.2

Simplify.

$y - 1 = {(x - 2)}^{3}$

Step 2.4.3

Simplify the right side.

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Step 2.4.3.1

Simplify ${(x - 2)}^{3}$ .

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Step 2.4.3.1.1

Use the Binomial Theorem.

$y - 1 = x^{3} + 3 x^{2} \cdot - 2 + 3 x {(- 2)}^{2} + {(- 2)}^{3}$

Step 2.4.3.1.2

Simplify each term.

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Step 2.4.3.1.2.1

Multiply $- 2$ by $3$ .

$y - 1 = x^{3} - 6 x^{2} + 3 x {(- 2)}^{2} + {(- 2)}^{3}$

Step 2.4.3.1.2.2

Raise $- 2$ to the power of $2$ .

$y - 1 = x^{3} - 6 x^{2} + 3 x \cdot 4 + {(- 2)}^{3}$

Step 2.4.3.1.2.3

Multiply $4$ by $3$ .

$y - 1 = x^{3} - 6 x^{2} + 12 x + {(- 2)}^{3}$

Step 2.4.3.1.2.4

Raise $- 2$ to the power of $3$ .

$y - 1 = x^{3} - 6 x^{2} + 12 x - 8$

Step 2.5

Move all terms not containing $y$ to the right side of the equation.

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Step 2.5.1

Add $1$ to both sides of the equation.

$y = x^{3} - 6 x^{2} + 12 x - 8 + 1$

Step 2.5.2

Add $- 8$ and $1$ .

$y = x^{3} - 6 x^{2} + 12 x - 7$

Step 3

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

$f^{- 1} (x) = x^{3} - 6 x^{2} + 12 x - 7$

Step 4

Verify if $f^{- 1} (x) = x^{3} - 6 x^{2} + 12 x - 7$ is the inverse of $f (x) = 2 + \sqrt[3]{x - 1}$ .

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Step 4.1

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

Step 4.2

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

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Step 4.2.1

Set up the composite result function.

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

Step 4.2.2

Evaluate $f^{- 1} (2 + \sqrt[3]{x - 1})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (2 + \sqrt[3]{x - 1}) = {(2 + \sqrt[3]{x - 1})}^{3} - 6 {(2 + \sqrt[3]{x - 1})}^{2} + 12 (2 + \sqrt[3]{x - 1}) - 7$

Step 4.2.3

Simplify each term.

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Step 4.2.3.1

Use the Binomial Theorem.

$f^{- 1} (2 + \sqrt[3]{x - 1}) = 2^{3} + 3 \cdot (2^{2} \sqrt[3]{x - 1}) + 3 \cdot (2 {\sqrt[3]{x - 1}}^{2}) + {\sqrt[3]{x - 1}}^{3} - 6 {(2 + \sqrt[3]{x - 1})}^{2} + 12 (2 + \sqrt[3]{x - 1}) - 7$

Step 4.2.3.2

Simplify each term.

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Step 4.2.3.2.1

Raise $2$ to the power of $3$ .

$f^{- 1} (2 + \sqrt[3]{x - 1}) = 8 + 3 \cdot (2^{2} \sqrt[3]{x - 1}) + 3 \cdot (2 {\sqrt[3]{x - 1}}^{2}) + {\sqrt[3]{x - 1}}^{3} - 6 {(2 + \sqrt[3]{x - 1})}^{2} + 12 (2 + \sqrt[3]{x - 1}) - 7$

Step 4.2.3.2.2

Raise $2$ to the power of $2$ .

$f^{- 1} (2 + \sqrt[3]{x - 1}) = 8 + 3 \cdot (4 \sqrt[3]{x - 1}) + 3 \cdot (2 {\sqrt[3]{x - 1}}^{2}) + {\sqrt[3]{x - 1}}^{3} - 6 {(2 + \sqrt[3]{x - 1})}^{2} + 12 (2 + \sqrt[3]{x - 1}) - 7$

Step 4.2.3.2.3

Multiply $3$ by $4$ .

$f^{- 1} (2 + \sqrt[3]{x - 1}) = 8 + 12 \sqrt[3]{x - 1} + 3 \cdot (2 {\sqrt[3]{x - 1}}^{2}) + {\sqrt[3]{x - 1}}^{3} - 6 {(2 + \sqrt[3]{x - 1})}^{2} + 12 (2 + \sqrt[3]{x - 1}) - 7$

Step 4.2.3.2.4

Multiply $3$ by $2$ .

$f^{- 1} (2 + \sqrt[3]{x - 1}) = 8 + 12 \sqrt[3]{x - 1} + 6 {\sqrt[3]{x - 1}}^{2} + {\sqrt[3]{x - 1}}^{3} - 6 {(2 + \sqrt[3]{x - 1})}^{2} + 12 (2 + \sqrt[3]{x - 1}) - 7$

Step 4.2.3.2.5

Rewrite ${\sqrt[3]{x - 1}}^{2}$ as $\sqrt[3]{{(x - 1)}^{2}}$ .

$f^{- 1} (2 + \sqrt[3]{x - 1}) = 8 + 12 \sqrt[3]{x - 1} + 6 \sqrt[3]{{(x - 1)}^{2}} + {\sqrt[3]{x - 1}}^{3} - 6 {(2 + \sqrt[3]{x - 1})}^{2} + 12 (2 + \sqrt[3]{x - 1}) - 7$

Step 4.2.3.2.6

Rewrite ${\sqrt[3]{x - 1}}^{3}$ as $x - 1$ .

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Step 4.2.3.2.6.1

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

$f^{- 1} (2 + \sqrt[3]{x - 1}) = 8 + 12 \sqrt[3]{x - 1} + 6 \sqrt[3]{{(x - 1)}^{2}} + {({(x - 1)}^{\frac{1}{3}})}^{3} - 6 {(2 + \sqrt[3]{x - 1})}^{2} + 12 (2 + \sqrt[3]{x - 1}) - 7$

Step 4.2.3.2.6.2

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

$f^{- 1} (2 + \sqrt[3]{x - 1}) = 8 + 12 \sqrt[3]{x - 1} + 6 \sqrt[3]{{(x - 1)}^{2}} + {(x - 1)}^{\frac{1}{3} \cdot 3} - 6 {(2 + \sqrt[3]{x - 1})}^{2} + 12 (2 + \sqrt[3]{x - 1}) - 7$

Step 4.2.3.2.6.3

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

$f^{- 1} (2 + \sqrt[3]{x - 1}) = 8 + 12 \sqrt[3]{x - 1} + 6 \sqrt[3]{{(x - 1)}^{2}} + {(x - 1)}^{\frac{3}{3}} - 6 {(2 + \sqrt[3]{x - 1})}^{2} + 12 (2 + \sqrt[3]{x - 1}) - 7$

Step 4.2.3.2.6.4

Cancel the common factor of $3$ .

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Step 4.2.3.2.6.4.1

Cancel the common factor.

$f^{- 1} (2 + \sqrt[3]{x - 1}) = 8 + 12 \sqrt[3]{x - 1} + 6 \sqrt[3]{{(x - 1)}^{2}} + {(x - 1)}^{\frac{3}{3}} - 6 {(2 + \sqrt[3]{x - 1})}^{2} + 12 (2 + \sqrt[3]{x - 1}) - 7$

Step 4.2.3.2.6.4.2

Rewrite the expression.

$f^{- 1} (2 + \sqrt[3]{x - 1}) = 8 + 12 \sqrt[3]{x - 1} + 6 \sqrt[3]{{(x - 1)}^{2}} + x - 1 - 6 {(2 + \sqrt[3]{x - 1})}^{2} + 12 (2 + \sqrt[3]{x - 1}) - 7$

Step 4.2.3.2.6.5

Simplify.

$f^{- 1} (2 + \sqrt[3]{x - 1}) = 8 + 12 \sqrt[3]{x - 1} + 6 \sqrt[3]{{(x - 1)}^{2}} + x - 1 - 6 {(2 + \sqrt[3]{x - 1})}^{2} + 12 (2 + \sqrt[3]{x - 1}) - 7$

Step 4.2.3.3

Subtract $1$ from $8$ .

$f^{- 1} (2 + \sqrt[3]{x - 1}) = 7 + 12 \sqrt[3]{x - 1} + 6 \sqrt[3]{{(x - 1)}^{2}} + x - 6 {(2 + \sqrt[3]{x - 1})}^{2} + 12 (2 + \sqrt[3]{x - 1}) - 7$

Step 4.2.3.4

Rewrite ${(2 + \sqrt[3]{x - 1})}^{2}$ as $(2 + \sqrt[3]{x - 1}) (2 + \sqrt[3]{x - 1})$ .

$f^{- 1} (2 + \sqrt[3]{x - 1}) = 7 + 12 \sqrt[3]{x - 1} + 6 \sqrt[3]{{(x - 1)}^{2}} + x - 6 ((2 + \sqrt[3]{x - 1}) (2 + \sqrt[3]{x - 1})) + 12 (2 + \sqrt[3]{x - 1}) - 7$

Step 4.2.3.5

Expand $(2 + \sqrt[3]{x - 1}) (2 + \sqrt[3]{x - 1})$ using the FOIL Method.

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Step 4.2.3.5.1

Apply the distributive property.

$f^{- 1} (2 + \sqrt[3]{x - 1}) = 7 + 12 \sqrt[3]{x - 1} + 6 \sqrt[3]{{(x - 1)}^{2}} + x - 6 (2 (2 + \sqrt[3]{x - 1}) + \sqrt[3]{x - 1} (2 + \sqrt[3]{x - 1})) + 12 (2 + \sqrt[3]{x - 1}) - 7$

Step 4.2.3.5.2

Apply the distributive property.

$f^{- 1} (2 + \sqrt[3]{x - 1}) = 7 + 12 \sqrt[3]{x - 1} + 6 \sqrt[3]{{(x - 1)}^{2}} + x - 6 (2 \cdot 2 + 2 \sqrt[3]{x - 1} + \sqrt[3]{x - 1} (2 + \sqrt[3]{x - 1})) + 12 (2 + \sqrt[3]{x - 1}) - 7$

Step 4.2.3.5.3

Apply the distributive property.

$f^{- 1} (2 + \sqrt[3]{x - 1}) = 7 + 12 \sqrt[3]{x - 1} + 6 \sqrt[3]{{(x - 1)}^{2}} + x - 6 (2 \cdot 2 + 2 \sqrt[3]{x - 1} + \sqrt[3]{x - 1} \cdot 2 + \sqrt[3]{x - 1} \sqrt[3]{x - 1}) + 12 (2 + \sqrt[3]{x - 1}) - 7$

Step 4.2.3.6

Simplify and combine like terms.

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Step 4.2.3.6.1

Simplify each term.

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Step 4.2.3.6.1.1

Multiply $2$ by $2$ .

$f^{- 1} (2 + \sqrt[3]{x - 1}) = 7 + 12 \sqrt[3]{x - 1} + 6 \sqrt[3]{{(x - 1)}^{2}} + x - 6 (4 + 2 \sqrt[3]{x - 1} + \sqrt[3]{x - 1} \cdot 2 + \sqrt[3]{x - 1} \sqrt[3]{x - 1}) + 12 (2 + \sqrt[3]{x - 1}) - 7$

Step 4.2.3.6.1.2

Move $2$ to the left of $\sqrt[3]{x - 1}$ .

$f^{- 1} (2 + \sqrt[3]{x - 1}) = 7 + 12 \sqrt[3]{x - 1} + 6 \sqrt[3]{{(x - 1)}^{2}} + x - 6 (4 + 2 \sqrt[3]{x - 1} + 2 \sqrt[3]{x - 1} + \sqrt[3]{x - 1} \sqrt[3]{x - 1}) + 12 (2 + \sqrt[3]{x - 1}) - 7$

Step 4.2.3.6.1.3

Multiply $\sqrt[3]{x - 1} \sqrt[3]{x - 1}$ .

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Step 4.2.3.6.1.3.1

Raise $\sqrt[3]{x - 1}$ to the power of $1$ .

Step 4.2.3.6.1.3.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f^{- 1} (2 + \sqrt[3]{x - 1}) = 7 + 12 \sqrt[3]{x - 1} + 6 \sqrt[3]{{(x - 1)}^{2}} + x - 6 (4 + 2 \sqrt[3]{x - 1} + 2 \sqrt[3]{x - 1} + {\sqrt[3]{x - 1}}^{1 + 1}) + 12 (2 + \sqrt[3]{x - 1}) - 7$

Step 4.2.3.6.1.3.3

Add $1$ and $1$ .

$f^{- 1} (2 + \sqrt[3]{x - 1}) = 7 + 12 \sqrt[3]{x - 1} + 6 \sqrt[3]{{(x - 1)}^{2}} + x - 6 (4 + 2 \sqrt[3]{x - 1} + 2 \sqrt[3]{x - 1} + {\sqrt[3]{x - 1}}^{2}) + 12 (2 + \sqrt[3]{x - 1}) - 7$

Step 4.2.3.6.1.4

Rewrite ${\sqrt[3]{x - 1}}^{2}$ as $\sqrt[3]{{(x - 1)}^{2}}$ .

$f^{- 1} (2 + \sqrt[3]{x - 1}) = 7 + 12 \sqrt[3]{x - 1} + 6 \sqrt[3]{{(x - 1)}^{2}} + x - 6 (4 + 2 \sqrt[3]{x - 1} + 2 \sqrt[3]{x - 1} + \sqrt[3]{{(x - 1)}^{2}}) + 12 (2 + \sqrt[3]{x - 1}) - 7$

Step 4.2.3.6.2

Add $2 \sqrt[3]{x - 1}$ and $2 \sqrt[3]{x - 1}$ .

$f^{- 1} (2 + \sqrt[3]{x - 1}) = 7 + 12 \sqrt[3]{x - 1} + 6 \sqrt[3]{{(x - 1)}^{2}} + x - 6 (4 + 4 \sqrt[3]{x - 1} + \sqrt[3]{{(x - 1)}^{2}}) + 12 (2 + \sqrt[3]{x - 1}) - 7$

Step 4.2.3.7

Apply the distributive property.

$f^{- 1} (2 + \sqrt[3]{x - 1}) = 7 + 12 \sqrt[3]{x - 1} + 6 \sqrt[3]{{(x - 1)}^{2}} + x - 6 \cdot 4 - 6 (4 \sqrt[3]{x - 1}) - 6 \sqrt[3]{{(x - 1)}^{2}} + 12 (2 + \sqrt[3]{x - 1}) - 7$

Step 4.2.3.8

Simplify.

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Step 4.2.3.8.1

Multiply $- 6$ by $4$ .

$f^{- 1} (2 + \sqrt[3]{x - 1}) = 7 + 12 \sqrt[3]{x - 1} + 6 \sqrt[3]{{(x - 1)}^{2}} + x - 24 - 6 (4 \sqrt[3]{x - 1}) - 6 \sqrt[3]{{(x - 1)}^{2}} + 12 (2 + \sqrt[3]{x - 1}) - 7$

Step 4.2.3.8.2

Multiply $4$ by $- 6$ .

$f^{- 1} (2 + \sqrt[3]{x - 1}) = 7 + 12 \sqrt[3]{x - 1} + 6 \sqrt[3]{{(x - 1)}^{2}} + x - 24 - 24 \sqrt[3]{x - 1} - 6 \sqrt[3]{{(x - 1)}^{2}} + 12 (2 + \sqrt[3]{x - 1}) - 7$

Step 4.2.3.9

Apply the distributive property.

$f^{- 1} (2 + \sqrt[3]{x - 1}) = 7 + 12 \sqrt[3]{x - 1} + 6 \sqrt[3]{{(x - 1)}^{2}} + x - 24 - 24 \sqrt[3]{x - 1} - 6 \sqrt[3]{{(x - 1)}^{2}} + 12 \cdot 2 + 12 \sqrt[3]{x - 1} - 7$

Step 4.2.3.10

Multiply $12$ by $2$ .

$f^{- 1} (2 + \sqrt[3]{x - 1}) = 7 + 12 \sqrt[3]{x - 1} + 6 \sqrt[3]{{(x - 1)}^{2}} + x - 24 - 24 \sqrt[3]{x - 1} - 6 \sqrt[3]{{(x - 1)}^{2}} + 24 + 12 \sqrt[3]{x - 1} - 7$

Step 4.2.4

Simplify by adding terms.

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Step 4.2.4.1

Combine the opposite terms in $7 + 12 \sqrt[3]{x - 1} + 6 \sqrt[3]{{(x - 1)}^{2}} + x - 24 - 24 \sqrt[3]{x - 1} - 6 \sqrt[3]{{(x - 1)}^{2}} + 24 + 12 \sqrt[3]{x - 1} - 7$ .

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Step 4.2.4.1.1

Subtract $6 \sqrt[3]{{(x - 1)}^{2}}$ from $6 \sqrt[3]{{(x - 1)}^{2}}$ .

$f^{- 1} (2 + \sqrt[3]{x - 1}) = 7 + 12 \sqrt[3]{x - 1} + 0 + x - 24 - 24 \sqrt[3]{x - 1} + 24 + 12 \sqrt[3]{x - 1} - 7$

Step 4.2.4.1.2

Add $7 + 12 \sqrt[3]{x - 1}$ and $0$ .

$f^{- 1} (2 + \sqrt[3]{x - 1}) = 7 + 12 \sqrt[3]{x - 1} + x - 24 - 24 \sqrt[3]{x - 1} + 24 + 12 \sqrt[3]{x - 1} - 7$

Step 4.2.4.1.3

Add $- 24$ and $24$ .

$f^{- 1} (2 + \sqrt[3]{x - 1}) = 7 + 12 \sqrt[3]{x - 1} + x + 0 - 24 \sqrt[3]{x - 1} + 12 \sqrt[3]{x - 1} - 7$

Step 4.2.4.1.4

Add $7 + 12 \sqrt[3]{x - 1} + x$ and $0$ .

$f^{- 1} (2 + \sqrt[3]{x - 1}) = 7 + 12 \sqrt[3]{x - 1} + x - 24 \sqrt[3]{x - 1} + 12 \sqrt[3]{x - 1} - 7$

Step 4.2.4.1.5

Subtract $7$ from $7$ .

$f^{- 1} (2 + \sqrt[3]{x - 1}) = 0 + 12 \sqrt[3]{x - 1} + x - 24 \sqrt[3]{x - 1} + 12 \sqrt[3]{x - 1}$

Step 4.2.4.1.6

Add $0$ and $12 \sqrt[3]{x - 1}$ .

$f^{- 1} (2 + \sqrt[3]{x - 1}) = 12 \sqrt[3]{x - 1} + x - 24 \sqrt[3]{x - 1} + 12 \sqrt[3]{x - 1}$

Step 4.2.4.2

Subtract $24 \sqrt[3]{x - 1}$ from $12 \sqrt[3]{x - 1}$ .

$f^{- 1} (2 + \sqrt[3]{x - 1}) = x - 12 \sqrt[3]{x - 1} + 12 \sqrt[3]{x - 1}$

Step 4.2.4.3

Combine the opposite terms in $x - 12 \sqrt[3]{x - 1} + 12 \sqrt[3]{x - 1}$ .

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Step 4.2.4.3.1

Add $- 12 \sqrt[3]{x - 1}$ and $12 \sqrt[3]{x - 1}$ .

$f^{- 1} (2 + \sqrt[3]{x - 1}) = x + 0$

Step 4.2.4.3.2

Add $x$ and $0$ .

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

Step 4.3

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

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Step 4.3.1

Set up the composite result function.

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

Step 4.3.2

Evaluate $f (x^{3} - 6 x^{2} + 12 x - 7)$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (x^{3} - 6 x^{2} + 12 x - 7) = 2 + \sqrt[3]{(x^{3} - 6 x^{2} + 12 x - 7) - 1}$

Step 4.3.3

Simplify each term.

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Step 4.3.3.1

Subtract $1$ from $- 7$ .

$f (x^{3} - 6 x^{2} + 12 x - 7) = 2 + \sqrt[3]{x^{3} - 6 x^{2} + 12 x - 8}$

Step 4.3.3.2

Rewrite $x^{3} - 6 x^{2} + 12 x - 8$ in a factored form.

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Step 4.3.3.2.1

Factor $x^{3} - 6 x^{2} + 12 x - 8$ using the rational roots test.

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Step 4.3.3.2.1.1

If a polynomial function has integer coefficients, then every rational zero will have the form $\frac{p}{q}$ where $p$ is a factor of the constant and $q$ is a factor of the leading coefficient.

$p = \pm 1, \pm 8, \pm 2, \pm 4$

$q = \pm 1$

Step 4.3.3.2.1.2

Find every combination of $\pm \frac{p}{q}$ . These are the possible roots of the polynomial function.

$\pm 1, \pm 8, \pm 2, \pm 4$

Step 4.3.3.2.1.3

Substitute $2$ and simplify the expression. In this case, the expression is equal to $0$ so $2$ is a root of the polynomial.

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Step 4.3.3.2.1.3.1

Substitute $2$ into the polynomial.

$2^{3} - 6 \cdot 2^{2} + 12 \cdot 2 - 8$

Step 4.3.3.2.1.3.2

Raise $2$ to the power of $3$ .

$8 - 6 \cdot 2^{2} + 12 \cdot 2 - 8$

Step 4.3.3.2.1.3.3

Raise $2$ to the power of $2$ .

$8 - 6 \cdot 4 + 12 \cdot 2 - 8$

Step 4.3.3.2.1.3.4

Multiply $- 6$ by $4$ .

$8 - 24 + 12 \cdot 2 - 8$

Step 4.3.3.2.1.3.5

Subtract $24$ from $8$ .

$- 16 + 12 \cdot 2 - 8$

Step 4.3.3.2.1.3.6

Multiply $12$ by $2$ .

$- 16 + 24 - 8$

Step 4.3.3.2.1.3.7

Add $- 16$ and $24$ .

$8 - 8$

Step 4.3.3.2.1.3.8

Subtract $8$ from $8$ .

$0$

Step 4.3.3.2.1.4

Since $2$ is a known root, divide the polynomial by $x - 2$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{x^{3} - 6 x^{2} + 12 x - 8}{x - 2}$

Step 4.3.3.2.1.5

Divide $x^{3} - 6 x^{2} + 12 x - 8$ by $x - 2$ .

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Step 4.3.3.2.1.5.1

Set up the polynomials to be divided. If there is not a term for every exponent, insert one with a value of $0$ .

$x$

$2$

$x^{3}$

$6 x^{2}$

$12 x$

$8$

Step 4.3.3.2.1.5.2

Divide the highest order term in the dividend $x^{3}$ by the highest order term in divisor $x$ .

					$x^{2}$
	$x$	-	$2$		$x^{3}$	-	$6 x^{2}$	+	$12 x$	-	$8$

Step 4.3.3.2.1.5.3

Multiply the new quotient term by the divisor.

				$x^{2}$
$x$	-	$2$		$x^{3}$	-	$6 x^{2}$	+	$12 x$	-	$8$
			+	$x^{3}$	-	$2 x^{2}$

Step 4.3.3.2.1.5.4

The expression needs to be subtracted from the dividend, so change all the signs in $x^{3} - 2 x^{2}$

				$x^{2}$
$x$	-	$2$		$x^{3}$	-	$6 x^{2}$	+	$12 x$	-	$8$
			-	$x^{3}$	+	$2 x^{2}$

Step 4.3.3.2.1.5.5

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

				$x^{2}$
$x$	-	$2$		$x^{3}$	-	$6 x^{2}$	+	$12 x$	-	$8$
			-	$x^{3}$	+	$2 x^{2}$
					-	$4 x^{2}$

Step 4.3.3.2.1.5.6

Pull the next terms from the original dividend down into the current dividend.

				$x^{2}$
$x$	-	$2$		$x^{3}$	-	$6 x^{2}$	+	$12 x$	-	$8$
			-	$x^{3}$	+	$2 x^{2}$
					-	$4 x^{2}$	+	$12 x$

Step 4.3.3.2.1.5.7

Divide the highest order term in the dividend $- 4 x^{2}$ by the highest order term in divisor $x$ .

				$x^{2}$	-	$4 x$
$x$	-	$2$		$x^{3}$	-	$6 x^{2}$	+	$12 x$	-	$8$
			-	$x^{3}$	+	$2 x^{2}$
					-	$4 x^{2}$	+	$12 x$

Step 4.3.3.2.1.5.8

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$4 x$
$x$	-	$2$		$x^{3}$	-	$6 x^{2}$	+	$12 x$	-	$8$
			-	$x^{3}$	+	$2 x^{2}$
					-	$4 x^{2}$	+	$12 x$
					-	$4 x^{2}$	+	$8 x$

Step 4.3.3.2.1.5.9

The expression needs to be subtracted from the dividend, so change all the signs in $- 4 x^{2} + 8 x$

				$x^{2}$	-	$4 x$
$x$	-	$2$		$x^{3}$	-	$6 x^{2}$	+	$12 x$	-	$8$
			-	$x^{3}$	+	$2 x^{2}$
					-	$4 x^{2}$	+	$12 x$
					+	$4 x^{2}$	-	$8 x$

Step 4.3.3.2.1.5.10

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

				$x^{2}$	-	$4 x$
$x$	-	$2$		$x^{3}$	-	$6 x^{2}$	+	$12 x$	-	$8$
			-	$x^{3}$	+	$2 x^{2}$
					-	$4 x^{2}$	+	$12 x$
					+	$4 x^{2}$	-	$8 x$
							+	$4 x$

Step 4.3.3.2.1.5.11

Pull the next terms from the original dividend down into the current dividend.

				$x^{2}$	-	$4 x$
$x$	-	$2$		$x^{3}$	-	$6 x^{2}$	+	$12 x$	-	$8$
			-	$x^{3}$	+	$2 x^{2}$
					-	$4 x^{2}$	+	$12 x$
					+	$4 x^{2}$	-	$8 x$
							+	$4 x$	-	$8$

Step 4.3.3.2.1.5.12

Divide the highest order term in the dividend $4 x$ by the highest order term in divisor $x$ .

				$x^{2}$	-	$4 x$	+	$4$
$x$	-	$2$		$x^{3}$	-	$6 x^{2}$	+	$12 x$	-	$8$
			-	$x^{3}$	+	$2 x^{2}$
					-	$4 x^{2}$	+	$12 x$
					+	$4 x^{2}$	-	$8 x$
							+	$4 x$	-	$8$

Step 4.3.3.2.1.5.13

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$4 x$	+	$4$
$x$	-	$2$		$x^{3}$	-	$6 x^{2}$	+	$12 x$	-	$8$
			-	$x^{3}$	+	$2 x^{2}$
					-	$4 x^{2}$	+	$12 x$
					+	$4 x^{2}$	-	$8 x$
							+	$4 x$	-	$8$
							+	$4 x$	-	$8$

Step 4.3.3.2.1.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $4 x - 8$

				$x^{2}$	-	$4 x$	+	$4$
$x$	-	$2$		$x^{3}$	-	$6 x^{2}$	+	$12 x$	-	$8$
			-	$x^{3}$	+	$2 x^{2}$
					-	$4 x^{2}$	+	$12 x$
					+	$4 x^{2}$	-	$8 x$
							+	$4 x$	-	$8$
							-	$4 x$	+	$8$

Step 4.3.3.2.1.5.15

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

				$x^{2}$	-	$4 x$	+	$4$
$x$	-	$2$		$x^{3}$	-	$6 x^{2}$	+	$12 x$	-	$8$
			-	$x^{3}$	+	$2 x^{2}$
					-	$4 x^{2}$	+	$12 x$
					+	$4 x^{2}$	-	$8 x$
							+	$4 x$	-	$8$
							-	$4 x$	+	$8$
										$0$

Step 4.3.3.2.1.5.16

Since the remander is $0$ , the final answer is the quotient.

$x^{2} - 4 x + 4$

Step 4.3.3.2.1.6

Write $x^{3} - 6 x^{2} + 12 x - 8$ as a set of factors.

$f (x^{3} - 6 x^{2} + 12 x - 7) = 2 + \sqrt[3]{(x - 2) (x^{2} - 4 x + 4)}$

Step 4.3.3.2.2

Factor using the perfect square rule.

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Step 4.3.3.2.2.1

Rewrite $4$ as $2^{2}$ .

$f (x^{3} - 6 x^{2} + 12 x - 7) = 2 + \sqrt[3]{(x - 2) (x^{2} - 4 x + 2^{2})}$

Step 4.3.3.2.2.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$4 x = 2 \cdot x \cdot 2$

Step 4.3.3.2.2.3

Rewrite the polynomial.

$f (x^{3} - 6 x^{2} + 12 x - 7) = 2 + \sqrt[3]{(x - 2) (x^{2} - 2 \cdot x \cdot 2 + 2^{2})}$

Step 4.3.3.2.2.4

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = x$ and $b = 2$ .

$f (x^{3} - 6 x^{2} + 12 x - 7) = 2 + \sqrt[3]{(x - 2) {(x - 2)}^{2}}$

Step 4.3.3.2.3

Combine like factors.

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Step 4.3.3.2.3.1

Raise $x - 2$ to the power of $1$ .

$f (x^{3} - 6 x^{2} + 12 x - 7) = 2 + \sqrt[3]{(x - 2) {(x - 2)}^{2}}$

Step 4.3.3.2.3.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f (x^{3} - 6 x^{2} + 12 x - 7) = 2 + \sqrt[3]{{(x - 2)}^{1 + 2}}$

Step 4.3.3.2.3.3

Add $1$ and $2$ .

$f (x^{3} - 6 x^{2} + 12 x - 7) = 2 + \sqrt[3]{{(x - 2)}^{3}}$

Step 4.3.3.3

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

$f (x^{3} - 6 x^{2} + 12 x - 7) = 2 + x - 2$

Step 4.3.4

Combine the opposite terms in $2 + x - 2$ .

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Step 4.3.4.1

Subtract $2$ from $2$ .

$f (x^{3} - 6 x^{2} + 12 x - 7) = x + 0$

Step 4.3.4.2

Add $x$ and $0$ .

$f (x^{3} - 6 x^{2} + 12 x - 7) = x$

Step 4.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = x^{3} - 6 x^{2} + 12 x - 7$ is the inverse of $f (x) = 2 + \sqrt[3]{x - 1}$ .

$f^{- 1} (x) = x^{3} - 6 x^{2} + 12 x - 7$