Algebra Examples

$- 2 x^{3} - 6$

Step 1

Interchange the variables.

$x = - 2 y^{3} - 6$

Step 2

Solve for $y$ .

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Step 2.1

Rewrite the equation as $- 2 y^{3} - 6 = x$ .

$- 2 y^{3} - 6 = x$

Step 2.2

Add $6$ to both sides of the equation.

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

Step 2.3

Divide each term in $- 2 y^{3} = x + 6$ by $- 2$ and simplify.

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Step 2.3.1

Divide each term in $- 2 y^{3} = x + 6$ by $- 2$ .

$\frac{- 2 y^{3}}{- 2} = \frac{x}{- 2} + \frac{6}{- 2}$

Step 2.3.2

Simplify the left side.

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Step 2.3.2.1

Cancel the common factor of $- 2$ .

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Step 2.3.2.1.1

Cancel the common factor.

$\frac{- 2 y^{3}}{- 2} = \frac{x}{- 2} + \frac{6}{- 2}$

Step 2.3.2.1.2

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

$y^{3} = \frac{x}{- 2} + \frac{6}{- 2}$

Step 2.3.3

Simplify the right side.

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Step 2.3.3.1

Simplify each term.

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Step 2.3.3.1.1

Move the negative in front of the fraction.

$y^{3} = - \frac{x}{2} + \frac{6}{- 2}$

Step 2.3.3.1.2

Divide $6$ by $- 2$ .

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

Step 2.4

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

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

Step 2.5

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

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Step 2.5.1

Factor $- 1$ out of $- \frac{x}{2} - 3$ .

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Step 2.5.1.1

Factor $- 1$ out of $- \frac{x}{2}$ .

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

Step 2.5.1.2

Rewrite $- 3$ as $- 1 (3)$ .

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

Step 2.5.1.3

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

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

Step 2.5.2

To write $3$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 2.5.3

Combine $3$ and $\frac{2}{2}$ .

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

Step 2.5.4

Combine the numerators over the common denominator.

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

Step 2.5.5

Multiply $3$ by $2$ .

$y = \sqrt[3]{- \frac{x + 6}{2}}$

Step 2.5.6

Rewrite $- \frac{x + 6}{2}$ as ${({(- 1)}^{3})}^{3} \frac{x + 6}{2}$ .

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Step 2.5.6.1

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

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

Step 2.5.6.2

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

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

Step 2.5.7

Pull terms out from under the radical.

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

Step 2.5.8

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

$y = - \sqrt[3]{\frac{x + 6}{2}}$

Step 2.5.9

Rewrite $\sqrt[3]{\frac{x + 6}{2}}$ as $\frac{\sqrt[3]{x + 6}}{\sqrt[3]{2}}$ .

$y = - \frac{\sqrt[3]{x + 6}}{\sqrt[3]{2}}$

Step 2.5.10

Multiply $\frac{\sqrt[3]{x + 6}}{\sqrt[3]{2}}$ by $\frac{{\sqrt[3]{2}}^{2}}{{\sqrt[3]{2}}^{2}}$ .

$y = - (\frac{\sqrt[3]{x + 6}}{\sqrt[3]{2}} \cdot \frac{{\sqrt[3]{2}}^{2}}{{\sqrt[3]{2}}^{2}})$

Step 2.5.11

Combine and simplify the denominator.

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Step 2.5.11.1

Multiply $\frac{\sqrt[3]{x + 6}}{\sqrt[3]{2}}$ by $\frac{{\sqrt[3]{2}}^{2}}{{\sqrt[3]{2}}^{2}}$ .

$y = - \frac{\sqrt[3]{x + 6} {\sqrt[3]{2}}^{2}}{\sqrt[3]{2} {\sqrt[3]{2}}^{2}}$

Step 2.5.11.2

Raise $\sqrt[3]{2}$ to the power of $1$ .

$y = - \frac{\sqrt[3]{x + 6} {\sqrt[3]{2}}^{2}}{{\sqrt[3]{2}}^{1} {\sqrt[3]{2}}^{2}}$

Step 2.5.11.3

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

$y = - \frac{\sqrt[3]{x + 6} {\sqrt[3]{2}}^{2}}{{\sqrt[3]{2}}^{1 + 2}}$

Step 2.5.11.4

Add $1$ and $2$ .

$y = - \frac{\sqrt[3]{x + 6} {\sqrt[3]{2}}^{2}}{{\sqrt[3]{2}}^{3}}$

Step 2.5.11.5

Rewrite ${\sqrt[3]{2}}^{3}$ as $2$ .

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Step 2.5.11.5.1

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

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

Step 2.5.11.5.2

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

$y = - \frac{\sqrt[3]{x + 6} {\sqrt[3]{2}}^{2}}{2^{\frac{1}{3} \cdot 3}}$

Step 2.5.11.5.3

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

$y = - \frac{\sqrt[3]{x + 6} {\sqrt[3]{2}}^{2}}{2^{\frac{3}{3}}}$

Step 2.5.11.5.4

Cancel the common factor of $3$ .

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Step 2.5.11.5.4.1

Cancel the common factor.

$y = - \frac{\sqrt[3]{x + 6} {\sqrt[3]{2}}^{2}}{2^{\frac{3}{3}}}$

Step 2.5.11.5.4.2

Rewrite the expression.

$y = - \frac{\sqrt[3]{x + 6} {\sqrt[3]{2}}^{2}}{2^{1}}$

Step 2.5.11.5.5

Evaluate the exponent.

$y = - \frac{\sqrt[3]{x + 6} {\sqrt[3]{2}}^{2}}{2}$

Step 2.5.12

Simplify the numerator.

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Step 2.5.12.1

Rewrite ${\sqrt[3]{2}}^{2}$ as $\sqrt[3]{2^{2}}$ .

$y = - \frac{\sqrt[3]{x + 6} \sqrt[3]{2^{2}}}{2}$

Step 2.5.12.2

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

$y = - \frac{\sqrt[3]{x + 6} \sqrt[3]{4}}{2}$

Step 2.5.13

Simplify with factoring out.

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Step 2.5.13.1

Combine using the product rule for radicals.

$y = - \frac{\sqrt[3]{(x + 6) \cdot 4}}{2}$

Step 2.5.13.2

Reorder factors in $- \frac{\sqrt[3]{(x + 6) \cdot 4}}{2}$ .

$y = - \frac{\sqrt[3]{4 (x + 6)}}{2}$

Step 3

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

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

Step 4

Verify if $f^{- 1} (x) = - \frac{\sqrt[3]{4 (x + 6)}}{2}$ is the inverse of $f (x) = - 2 x^{3} - 6$ .

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

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

Step 4.2.3

Simplify the numerator.

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Step 4.2.3.1

Add $- 6$ and $6$ .

$f^{- 1} (- 2 x^{3} - 6) = - \frac{\sqrt[3]{4 (- 2 x^{3} + 0)}}{2}$

Step 4.2.3.2

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

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

Step 4.2.3.3

Multiply $4$ by $- 2$ .

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

Step 4.2.3.4

Rewrite $- 8 x^{3}$ as ${(- 2 x)}^{3}$ .

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

Step 4.2.3.5

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

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

Step 4.2.4

Cancel the common factor of $- 2$ and $2$ .

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Step 4.2.4.1

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

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

Step 4.2.4.2

Cancel the common factors.

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Step 4.2.4.2.1

Factor $2$ out of $2$ .

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

Step 4.2.4.2.2

Cancel the common factor.

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

Step 4.2.4.2.3

Rewrite the expression.

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

Step 4.2.4.2.4

Divide $- x$ by $1$ .

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

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 $f (- \frac{\sqrt[3]{4 (x + 6)}}{2})$ by substituting in the value of $f^{- 1}$ into $f$ .

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

Step 4.3.3

Simplify each term.

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Step 4.3.3.1

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 4.3.3.1.1

Apply the product rule to $- \frac{\sqrt[3]{4 (x + 6)}}{2}$ .

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

Step 4.3.3.1.2

Apply the product rule to $\frac{\sqrt[3]{4 (x + 6)}}{2}$ .

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

Step 4.3.3.2

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

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

Step 4.3.3.3

Simplify the numerator.

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Step 4.3.3.3.1

Rewrite ${\sqrt[3]{4 (x + 6)}}^{3}$ as $4 (x + 6)$ .

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Step 4.3.3.3.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{4 (x + 6)}$ as ${(4 (x + 6))}^{\frac{1}{3}}$ .

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

Step 4.3.3.3.1.2

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

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

Step 4.3.3.3.1.3

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

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

Step 4.3.3.3.1.4

Cancel the common factor of $3$ .

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Step 4.3.3.3.1.4.1

Cancel the common factor.

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

Step 4.3.3.3.1.4.2

Rewrite the expression.

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

Step 4.3.3.3.1.5

Simplify.

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

Step 4.3.3.3.2

Apply the distributive property.

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

Step 4.3.3.3.3

Multiply $4$ by $6$ .

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

Step 4.3.3.3.4

Factor $4$ out of $4 x + 24$ .

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Step 4.3.3.3.4.1

Factor $4$ out of $4 x$ .

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

Step 4.3.3.3.4.2

Factor $4$ out of $24$ .

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

Step 4.3.3.3.4.3

Factor $4$ out of $4 x + 4 \cdot 6$ .

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

Step 4.3.3.4

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

$f (- \frac{\sqrt[3]{4 (x + 6)}}{2}) = - 2 (- \frac{4 (x + 6)}{8}) - 6$

Step 4.3.3.5

Cancel the common factor of $2$ .

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Step 4.3.3.5.1

Move the leading negative in $- \frac{4 (x + 6)}{8}$ into the numerator.

$f (- \frac{\sqrt[3]{4 (x + 6)}}{2}) = - 2 \frac{- 4 (x + 6)}{8} - 6$

Step 4.3.3.5.2

Factor $2$ out of $- 2$ .

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

Step 4.3.3.5.3

Factor $2$ out of $8$ .

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

Step 4.3.3.5.4

Cancel the common factor.

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

Step 4.3.3.5.5

Rewrite the expression.

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

Step 4.3.3.6

Cancel the common factor of $- 4$ and $4$ .

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Step 4.3.3.6.1

Factor $4$ out of $- 4 (x + 6)$ .

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

Step 4.3.3.6.2

Cancel the common factors.

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Step 4.3.3.6.2.1

Factor $4$ out of $4$ .

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

Step 4.3.3.6.2.2

Cancel the common factor.

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

Step 4.3.3.6.2.3

Rewrite the expression.

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

Step 4.3.3.6.2.4

Divide $- (x + 6)$ by $1$ .

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

Step 4.3.3.7

Apply the distributive property.

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

Step 4.3.3.8

Multiply $- 1$ by $6$ .

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

Step 4.3.3.9

Apply the distributive property.

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

Step 4.3.3.10

Multiply $- 1 (- x)$ .

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Step 4.3.3.10.1

Multiply $- 1$ by $- 1$ .

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

Step 4.3.3.10.2

Multiply $x$ by $1$ .

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

Step 4.3.3.11

Multiply $- 1$ by $- 6$ .

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

Step 4.3.4

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

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Step 4.3.4.1

Subtract $6$ from $6$ .

$f (- \frac{\sqrt[3]{4 (x + 6)}}{2}) = x + 0$

Step 4.3.4.2

Add $x$ and $0$ .

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

Step 4.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = - \frac{\sqrt[3]{4 (x + 6)}}{2}$ is the inverse of $f (x) = - 2 x^{3} - 6$ .

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