Algebra Examples

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

Step 1

Write $f (x) = 1 - \frac{2}{x^{3}}$ as an equation.

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

Step 2

Interchange the variables.

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

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $1 - \frac{2}{y^{3}} = x$ .

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

Step 3.2

Subtract $1$ from both sides of the equation.

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

Step 3.3

Find the LCD of the terms in the equation.

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Step 3.3.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$y^{3}, 1, 1$

Step 3.3.2

The LCM of one and any expression is the expression.

$y^{3}$

Step 3.4

Multiply each term in $- \frac{2}{y^{3}} = x - 1$ by $y^{3}$ to eliminate the fractions.

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Step 3.4.1

Multiply each term in $- \frac{2}{y^{3}} = x - 1$ by $y^{3}$ .

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

Step 3.4.2

Simplify the left side.

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Step 3.4.2.1

Cancel the common factor of $y^{3}$ .

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Step 3.4.2.1.1

Move the leading negative in $- \frac{2}{y^{3}}$ into the numerator.

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

Step 3.4.2.1.2

Cancel the common factor.

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

Step 3.4.2.1.3

Rewrite the expression.

$- 2 = x y^{3} - y^{3}$

Step 3.5

Solve the equation.

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Step 3.5.1

Rewrite the equation as $x y^{3} - y^{3} = - 2$ .

$x y^{3} - y^{3} = - 2$

Step 3.5.2

Factor $y^{3}$ out of $x y^{3} - y^{3}$ .

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Step 3.5.2.1

Factor $y^{3}$ out of $x y^{3}$ .

$y^{3} x - y^{3} = - 2$

Step 3.5.2.2

Factor $y^{3}$ out of $- y^{3}$ .

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

Step 3.5.2.3

Factor $y^{3}$ out of $y^{3} x + y^{3} \cdot - 1$ .

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

Step 3.5.3

Divide each term in $y^{3} (x - 1) = - 2$ by $x - 1$ and simplify.

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Step 3.5.3.1

Divide each term in $y^{3} (x - 1) = - 2$ by $x - 1$ .

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

Step 3.5.3.2

Simplify the left side.

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Step 3.5.3.2.1

Cancel the common factor of $x - 1$ .

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Step 3.5.3.2.1.1

Cancel the common factor.

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

Step 3.5.3.2.1.2

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

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

Step 3.5.3.3

Simplify the right side.

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Step 3.5.3.3.1

Move the negative in front of the fraction.

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

Step 3.5.4

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

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

Step 3.5.5

Simplify $\sqrt[3]{- \frac{2}{x - 1}}$ .

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Step 3.5.5.1

Rewrite $- \frac{2}{x - 1}$ as ${({(- 1)}^{3})}^{3} \frac{2}{x - 1}$ .

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Step 3.5.5.1.1

Rewrite $- 1$ as ${(- 1)}^{3}$ .

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

Step 3.5.5.1.2

Rewrite $- 1$ as ${(- 1)}^{3}$ .

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

Step 3.5.5.2

Pull terms out from under the radical.

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

Step 3.5.5.3

Raise $- 1$ to the power of $3$ .

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

Step 3.5.5.4

Rewrite $\sqrt[3]{\frac{2}{x - 1}}$ as $\frac{\sqrt[3]{2}}{\sqrt[3]{x - 1}}$ .

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

Step 3.5.5.5

Multiply $\frac{\sqrt[3]{2}}{\sqrt[3]{x - 1}}$ by $\frac{{\sqrt[3]{x - 1}}^{2}}{{\sqrt[3]{x - 1}}^{2}}$ .

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

Step 3.5.5.6

Combine and simplify the denominator.

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Step 3.5.5.6.1

Multiply $\frac{\sqrt[3]{2}}{\sqrt[3]{x - 1}}$ by $\frac{{\sqrt[3]{x - 1}}^{2}}{{\sqrt[3]{x - 1}}^{2}}$ .

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

Step 3.5.5.6.2

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

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

Step 3.5.5.6.3

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

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

Step 3.5.5.6.4

Add $1$ and $2$ .

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

Step 3.5.5.6.5

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

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Step 3.5.5.6.5.1

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

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

Step 3.5.5.6.5.2

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

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

Step 3.5.5.6.5.3

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

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

Step 3.5.5.6.5.4

Cancel the common factor of $3$ .

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Step 3.5.5.6.5.4.1

Cancel the common factor.

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

Step 3.5.5.6.5.4.2

Rewrite the expression.

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

Step 3.5.5.6.5.5

Simplify.

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

Step 3.5.5.7

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

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

Step 3.5.5.8

Combine using the product rule for radicals.

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

Step 4

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

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

Step 5

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

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Step 5.1

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

Step 5.2

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

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Step 5.2.1

Set up the composite result function.

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

Step 5.2.2

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

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

Step 5.2.3

Simplify the numerator.

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Step 5.2.3.1

Write $1$ as a fraction with a common denominator.

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

Step 5.2.3.2

Combine the numerators over the common denominator.

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

Step 5.2.3.3

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{x^{3}}{x^{3}}$ .

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

Step 5.2.3.4

Combine $- 1$ and $\frac{x^{3}}{x^{3}}$ .

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

Step 5.2.3.5

Combine the numerators over the common denominator.

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

Step 5.2.3.6

Reorder terms.

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

Step 5.2.3.7

Rewrite $\frac{x^{3} - x^{3} - 2}{x^{3}}$ in a factored form.

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Step 5.2.3.7.1

Subtract $x^{3}$ from $x^{3}$ .

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

Step 5.2.3.7.2

Subtract $2$ from $0$ .

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

Step 5.2.3.8

Apply the product rule to $\frac{- 2}{x^{3}}$ .

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

Step 5.2.3.9

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

$f^{- 1} (1 - \frac{2}{x^{3}}) = - \frac{\sqrt[3]{2 (\frac{4}{{(x^{3})}^{2}})}}{1 - \frac{2}{x^{3}} - 1}$

Step 5.2.3.10

Multiply the exponents in ${(x^{3})}^{2}$ .

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Step 5.2.3.10.1

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

$f^{- 1} (1 - \frac{2}{x^{3}}) = - \frac{\sqrt[3]{2 (\frac{4}{x^{3 \cdot 2}})}}{1 - \frac{2}{x^{3}} - 1}$

Step 5.2.3.10.2

Multiply $3$ by $2$ .

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

Step 5.2.3.11

Combine $2$ and $\frac{4}{x^{6}}$ .

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

Step 5.2.3.12

Multiply $2$ by $4$ .

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

Step 5.2.3.13

Rewrite $8$ as $2^{3}$ .

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

Step 5.2.3.14

Rewrite $x^{6}$ as ${(x^{2})}^{3}$ .

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

Step 5.2.3.15

Rewrite $\frac{2^{3}}{{(x^{2})}^{3}}$ as ${(\frac{2}{x^{2}})}^{3}$ .

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

Step 5.2.3.16

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

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

Step 5.2.4

Simplify the denominator.

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Step 5.2.4.1

Subtract $1$ from $1$ .

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

Step 5.2.4.2

Add $- \frac{2}{x^{3}}$ and $0$ .

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

Step 5.2.5

Multiply the numerator by the reciprocal of the denominator.

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

Step 5.2.6

Rewrite using the commutative property of multiplication.

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

Step 5.2.7

Cancel the common factor of $2$ .

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Step 5.2.7.1

Move the leading negative in $- \frac{2}{x^{2}}$ into the numerator.

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

Step 5.2.7.2

Factor $2$ out of $- 2$ .

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

Step 5.2.7.3

Cancel the common factor.

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

Step 5.2.7.4

Rewrite the expression.

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

Step 5.2.8

Cancel the common factor of $x^{2}$ .

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Step 5.2.8.1

Factor $x^{2}$ out of $x^{3}$ .

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

Step 5.2.8.2

Cancel the common factor.

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

Step 5.2.8.3

Rewrite the expression.

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

Step 5.3

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

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Step 5.3.1

Set up the composite result function.

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

Step 5.3.2

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

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

Step 5.3.3

Simplify each term.

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Step 5.3.3.1

Simplify the denominator.

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Step 5.3.3.1.1

Apply the product rule to $- \frac{\sqrt[3]{2 {(x - 1)}^{2}}}{x - 1}$ .

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

Step 5.3.3.1.2

Raise $- 1$ to the power of $3$ .

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

Step 5.3.3.1.3

Apply the product rule to $\frac{\sqrt[3]{2 {(x - 1)}^{2}}}{x - 1}$ .

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

Step 5.3.3.1.4

Simplify the numerator.

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Step 5.3.3.1.4.1

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

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Step 5.3.3.1.4.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{2 {(x - 1)}^{2}}$ as ${(2 {(x - 1)}^{2})}^{\frac{1}{3}}$ .

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

Step 5.3.3.1.4.1.2

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

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

Step 5.3.3.1.4.1.3

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

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

Step 5.3.3.1.4.1.4

Cancel the common factor of $3$ .

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Step 5.3.3.1.4.1.4.1

Cancel the common factor.

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

Step 5.3.3.1.4.1.4.2

Rewrite the expression.

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

Step 5.3.3.1.4.1.5

Simplify.

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

Step 5.3.3.1.4.2

Rewrite ${(x - 1)}^{2}$ as $(x - 1) (x - 1)$ .

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

Step 5.3.3.1.4.3

Expand $(x - 1) (x - 1)$ using the FOIL Method.

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Step 5.3.3.1.4.3.1

Apply the distributive property.

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

Step 5.3.3.1.4.3.2

Apply the distributive property.

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

Step 5.3.3.1.4.3.3

Apply the distributive property.

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

Step 5.3.3.1.4.4

Simplify and combine like terms.

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Step 5.3.3.1.4.4.1

Simplify each term.

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Step 5.3.3.1.4.4.1.1

Multiply $x$ by $x$ .

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

Step 5.3.3.1.4.4.1.2

Move $- 1$ to the left of $x$ .

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

Step 5.3.3.1.4.4.1.3

Rewrite $- 1 x$ as $- x$ .

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

Step 5.3.3.1.4.4.1.4

Rewrite $- 1 x$ as $- x$ .

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

Step 5.3.3.1.4.4.1.5

Multiply $- 1$ by $- 1$ .

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

Step 5.3.3.1.4.4.2

Subtract $x$ from $- x$ .

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

Step 5.3.3.1.4.5

Apply the distributive property.

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

Step 5.3.3.1.4.6

Simplify.

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Step 5.3.3.1.4.6.1

Multiply $- 2$ by $2$ .

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

Step 5.3.3.1.4.6.2

Multiply $2$ by $1$ .

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

Step 5.3.3.1.4.7

Factor $2$ out of $2 x^{2} - 4 x + 2$ .

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Step 5.3.3.1.4.7.1

Factor $2$ out of $2 x^{2}$ .

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

Step 5.3.3.1.4.7.2

Factor $2$ out of $- 4 x$ .

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

Step 5.3.3.1.4.7.3

Factor $2$ out of $2$ .

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

Step 5.3.3.1.4.7.4

Factor $2$ out of $2 (x^{2}) + 2 (- 2 x)$ .

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

Step 5.3.3.1.4.7.5

Factor $2$ out of $2 (x^{2} - 2 x) + 2 (1)$ .

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

Step 5.3.3.1.4.8

Factor using the perfect square rule.

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Step 5.3.3.1.4.8.1

Rewrite $1$ as $1^{2}$ .

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

Step 5.3.3.1.4.8.2

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

$2 x = 2 \cdot x \cdot 1$

Step 5.3.3.1.4.8.3

Rewrite the polynomial.

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

Step 5.3.3.1.4.8.4

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = x$ and $b = 1$ .

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

Step 5.3.3.1.5

Cancel the common factor of ${(x - 1)}^{2}$ and ${(x - 1)}^{3}$ .

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Step 5.3.3.1.5.1

Factor ${(x - 1)}^{2}$ out of $2 {(x - 1)}^{2}$ .

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

Step 5.3.3.1.5.2

Cancel the common factors.

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Step 5.3.3.1.5.2.1

Factor ${(x - 1)}^{2}$ out of ${(x - 1)}^{3}$ .

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

Step 5.3.3.1.5.2.2

Cancel the common factor.

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

Step 5.3.3.1.5.2.3

Rewrite the expression.

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

Step 5.3.3.2

Multiply the numerator by the reciprocal of the denominator.

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

Step 5.3.3.3

Cancel the common factor of $2$ .

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Step 5.3.3.3.1

Move the leading negative in $- \frac{x - 1}{2}$ into the numerator.

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

Step 5.3.3.3.2

Cancel the common factor.

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

Step 5.3.3.3.3

Rewrite the expression.

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

Step 5.3.3.4

Apply the distributive property.

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

Step 5.3.3.5

Multiply $- 1$ by $- 1$ .

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

Step 5.3.3.6

Apply the distributive property.

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

Step 5.3.3.7

Multiply $- - x$ .

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Step 5.3.3.7.1

Multiply $- 1$ by $- 1$ .

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

Step 5.3.3.7.2

Multiply $x$ by $1$ .

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

Step 5.3.3.8

Multiply $- 1$ by $1$ .

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

Step 5.3.4

Combine the opposite terms in $1 + x - 1$ .

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Step 5.3.4.1

Subtract $1$ from $1$ .

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

Step 5.3.4.2

Add $x$ and $0$ .

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

Step 5.4

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

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