Algebra Examples

$9 - x^{3}$

Step 1

Interchange the variables.

$x = 9 - y^{3}$

Step 2

Solve for $y$ .

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Step 2.1

Rewrite the equation as $9 - y^{3} = x$ .

$9 - y^{3} = x$

Step 2.2

Subtract $9$ from both sides of the equation.

$- y^{3} = x - 9$

Step 2.3

Divide each term in $- y^{3} = x - 9$ by $- 1$ and simplify.

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Step 2.3.1

Divide each term in $- y^{3} = x - 9$ by $- 1$ .

$\frac{- y^{3}}{- 1} = \frac{x}{- 1} + \frac{- 9}{- 1}$

Step 2.3.2

Simplify the left side.

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Step 2.3.2.1

Dividing two negative values results in a positive value.

$\frac{y^{3}}{1} = \frac{x}{- 1} + \frac{- 9}{- 1}$

Step 2.3.2.2

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

$y^{3} = \frac{x}{- 1} + \frac{- 9}{- 1}$

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

$y^{3} = - 1 \cdot x + \frac{- 9}{- 1}$

Step 2.3.3.1.2

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

$y^{3} = - x + \frac{- 9}{- 1}$

Step 2.3.3.1.3

Divide $- 9$ by $- 1$ .

$y^{3} = - x + 9$

Step 2.4

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

$y = \sqrt[3]{- x + 9}$

Step 3

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

$f^{- 1} (x) = \sqrt[3]{- x + 9}$

Step 4

Verify if $f^{- 1} (x) = \sqrt[3]{- x + 9}$ is the inverse of $f (x) = 9 - x^{3}$ .

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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 $f^{- 1} (9 - x^{3})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (9 - x^{3}) = \sqrt[3]{- (9 - x^{3}) + 9}$

Step 4.2.3

Apply the distributive property.

$f^{- 1} (9 - x^{3}) = \sqrt[3]{- 1 \cdot 9 + x^{3} + 9}$

Step 4.2.4

Multiply $- 1$ by $9$ .

$f^{- 1} (9 - x^{3}) = \sqrt[3]{- 9 + x^{3} + 9}$

Step 4.2.5

Add $- 9$ and $9$ .

$f^{- 1} (9 - x^{3}) = \sqrt[3]{x^{3} + 0}$

Step 4.2.6

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

$f^{- 1} (9 - x^{3}) = \sqrt[3]{x^{3}}$

Step 4.2.7

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

$f^{- 1} (9 - x^{3}) = x$

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 $f (\sqrt[3]{- x + 9})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\sqrt[3]{- x + 9}) = 9 - {(\sqrt[3]{- x + 9})}^{3}$

Step 4.3.3

Simplify each term.

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Step 4.3.3.1

Rewrite ${\sqrt[3]{- x + 9}}^{3}$ as $- x + 9$ .

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Step 4.3.3.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{- x + 9}$ as ${(- x + 9)}^{\frac{1}{3}}$ .

$f (\sqrt[3]{- x + 9}) = 9 - {({(- x + 9)}^{\frac{1}{3}})}^{3}$

Step 4.3.3.1.2

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

$f (\sqrt[3]{- x + 9}) = 9 - {(- x + 9)}^{\frac{1}{3} \cdot 3}$

Step 4.3.3.1.3

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

$f (\sqrt[3]{- x + 9}) = 9 - {(- x + 9)}^{\frac{3}{3}}$

Step 4.3.3.1.4

Cancel the common factor of $3$ .

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Step 4.3.3.1.4.1

Cancel the common factor.

$f (\sqrt[3]{- x + 9}) = 9 - {(- x + 9)}^{\frac{3}{3}}$

Step 4.3.3.1.4.2

Rewrite the expression.

$f (\sqrt[3]{- x + 9}) = 9 - (- x + 9)$

Step 4.3.3.1.5

Simplify.

$f (\sqrt[3]{- x + 9}) = 9 - (- x + 9)$

Step 4.3.3.2

Apply the distributive property.

$f (\sqrt[3]{- x + 9}) = 9 + x - 1 \cdot 9$

Step 4.3.3.3

Multiply $- - x$ .

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Step 4.3.3.3.1

Multiply $- 1$ by $- 1$ .

$f (\sqrt[3]{- x + 9}) = 9 + 1 x - 1 \cdot 9$

Step 4.3.3.3.2

Multiply $x$ by $1$ .

$f (\sqrt[3]{- x + 9}) = 9 + x - 1 \cdot 9$

Step 4.3.3.4

Multiply $- 1$ by $9$ .

$f (\sqrt[3]{- x + 9}) = 9 + x - 9$

Step 4.3.4

Combine the opposite terms in $9 + x - 9$ .

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Step 4.3.4.1

Subtract $9$ from $9$ .

$f (\sqrt[3]{- x + 9}) = x + 0$

Step 4.3.4.2

Add $x$ and $0$ .

$f (\sqrt[3]{- x + 9}) = x$

Step 4.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \sqrt[3]{- x + 9}$ is the inverse of $f (x) = 9 - x^{3}$ .

$f^{- 1} (x) = \sqrt[3]{- x + 9}$