Algebra Examples

$y = 3 x^{3} - 5$

Step 1

Interchange the variables.

$x = 3 y^{3} - 5$

Step 2

Solve for $y$ .

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Step 2.1

Rewrite the equation as $3 y^{3} - 5 = x$ .

$3 y^{3} - 5 = x$

Step 2.2

Add $5$ to both sides of the equation.

$3 y^{3} = x + 5$

Step 2.3

Divide each term in $3 y^{3} = x + 5$ by $3$ and simplify.

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Step 2.3.1

Divide each term in $3 y^{3} = x + 5$ by $3$ .

$\frac{3 y^{3}}{3} = \frac{x}{3} + \frac{5}{3}$

Step 2.3.2

Simplify the left side.

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Step 2.3.2.1

Cancel the common factor of $3$ .

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Step 2.3.2.1.1

Cancel the common factor.

$\frac{3 y^{3}}{3} = \frac{x}{3} + \frac{5}{3}$

Step 2.3.2.1.2

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

$y^{3} = \frac{x}{3} + \frac{5}{3}$

Step 2.4

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

$y = \sqrt[3]{\frac{x}{3} + \frac{5}{3}}$

Step 2.5

Simplify $\sqrt[3]{\frac{x}{3} + \frac{5}{3}}$ .

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Step 2.5.1

Combine the numerators over the common denominator.

$y = \sqrt[3]{\frac{x + 5}{3}}$

Step 2.5.2

Rewrite $\sqrt[3]{\frac{x + 5}{3}}$ as $\frac{\sqrt[3]{x + 5}}{\sqrt[3]{3}}$ .

$y = \frac{\sqrt[3]{x + 5}}{\sqrt[3]{3}}$

Step 2.5.3

Multiply $\frac{\sqrt[3]{x + 5}}{\sqrt[3]{3}}$ by $\frac{{\sqrt[3]{3}}^{2}}{{\sqrt[3]{3}}^{2}}$ .

$y = \frac{\sqrt[3]{x + 5}}{\sqrt[3]{3}} \cdot \frac{{\sqrt[3]{3}}^{2}}{{\sqrt[3]{3}}^{2}}$

Step 2.5.4

Combine and simplify the denominator.

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Step 2.5.4.1

Multiply $\frac{\sqrt[3]{x + 5}}{\sqrt[3]{3}}$ by $\frac{{\sqrt[3]{3}}^{2}}{{\sqrt[3]{3}}^{2}}$ .

$y = \frac{\sqrt[3]{x + 5} {\sqrt[3]{3}}^{2}}{\sqrt[3]{3} {\sqrt[3]{3}}^{2}}$

Step 2.5.4.2

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

$y = \frac{\sqrt[3]{x + 5} {\sqrt[3]{3}}^{2}}{{\sqrt[3]{3}}^{1} {\sqrt[3]{3}}^{2}}$

Step 2.5.4.3

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

$y = \frac{\sqrt[3]{x + 5} {\sqrt[3]{3}}^{2}}{{\sqrt[3]{3}}^{1 + 2}}$

Step 2.5.4.4

Add $1$ and $2$ .

$y = \frac{\sqrt[3]{x + 5} {\sqrt[3]{3}}^{2}}{{\sqrt[3]{3}}^{3}}$

Step 2.5.4.5

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

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Step 2.5.4.5.1

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

$y = \frac{\sqrt[3]{x + 5} {\sqrt[3]{3}}^{2}}{{(3^{\frac{1}{3}})}^{3}}$

Step 2.5.4.5.2

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

$y = \frac{\sqrt[3]{x + 5} {\sqrt[3]{3}}^{2}}{3^{\frac{1}{3} \cdot 3}}$

Step 2.5.4.5.3

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

$y = \frac{\sqrt[3]{x + 5} {\sqrt[3]{3}}^{2}}{3^{\frac{3}{3}}}$

Step 2.5.4.5.4

Cancel the common factor of $3$ .

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Step 2.5.4.5.4.1

Cancel the common factor.

$y = \frac{\sqrt[3]{x + 5} {\sqrt[3]{3}}^{2}}{3^{\frac{3}{3}}}$

Step 2.5.4.5.4.2

Rewrite the expression.

$y = \frac{\sqrt[3]{x + 5} {\sqrt[3]{3}}^{2}}{3^{1}}$

Step 2.5.4.5.5

Evaluate the exponent.

$y = \frac{\sqrt[3]{x + 5} {\sqrt[3]{3}}^{2}}{3}$

Step 2.5.5

Simplify the numerator.

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Step 2.5.5.1

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

$y = \frac{\sqrt[3]{x + 5} \sqrt[3]{3^{2}}}{3}$

Step 2.5.5.2

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

$y = \frac{\sqrt[3]{x + 5} \sqrt[3]{9}}{3}$

Step 2.5.6

Simplify with factoring out.

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Step 2.5.6.1

Combine using the product rule for radicals.

$y = \frac{\sqrt[3]{(x + 5) \cdot 9}}{3}$

Step 2.5.6.2

Reorder factors in $\frac{\sqrt[3]{(x + 5) \cdot 9}}{3}$ .

$y = \frac{\sqrt[3]{9 (x + 5)}}{3}$

Step 3

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

$f^{- 1} (x) = \frac{\sqrt[3]{9 (x + 5)}}{3}$

Step 4

Verify if $f^{- 1} (x) = \frac{\sqrt[3]{9 (x + 5)}}{3}$ is the inverse of $f (x) = 3 x^{3} - 5$ .

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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 $f^{- 1} (3 x^{3} - 5)$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (3 x^{3} - 5) = \frac{\sqrt[3]{9 ((3 x^{3} - 5) + 5)}}{3}$

Step 4.2.3

Simplify the numerator.

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Step 4.2.3.1

Add $- 5$ and $5$ .

$f^{- 1} (3 x^{3} - 5) = \frac{\sqrt[3]{9 (3 x^{3} + 0)}}{3}$

Step 4.2.3.2

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

$f^{- 1} (3 x^{3} - 5) = \frac{\sqrt[3]{9 \cdot (3 x^{3})}}{3}$

Step 4.2.3.3

Multiply $9$ by $3$ .

$f^{- 1} (3 x^{3} - 5) = \frac{\sqrt[3]{27 x^{3}}}{3}$

Step 4.2.3.4

Rewrite $27 x^{3}$ as ${(3 x)}^{3}$ .

$f^{- 1} (3 x^{3} - 5) = \frac{\sqrt[3]{{(3 x)}^{3}}}{3}$

Step 4.2.3.5

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

$f^{- 1} (3 x^{3} - 5) = \frac{3 x}{3}$

Step 4.2.4

Cancel the common factor of $3$ .

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Step 4.2.4.1

Cancel the common factor.

$f^{- 1} (3 x^{3} - 5) = \frac{3 x}{3}$

Step 4.2.4.2

Divide $x$ by $1$ .

$f^{- 1} (3 x^{3} - 5) = x$

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 $f (\frac{\sqrt[3]{9 (x + 5)}}{3})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\frac{\sqrt[3]{9 (x + 5)}}{3}) = 3 {(\frac{\sqrt[3]{9 (x + 5)}}{3})}^{3} - 5$

Step 4.3.3

Simplify each term.

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Step 4.3.3.1

Apply the product rule to $\frac{\sqrt[3]{9 (x + 5)}}{3}$ .

$f (\frac{\sqrt[3]{9 (x + 5)}}{3}) = 3 (\frac{{\sqrt[3]{9 (x + 5)}}^{3}}{3^{3}}) - 5$

Step 4.3.3.2

Simplify the numerator.

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Step 4.3.3.2.1

Rewrite ${\sqrt[3]{9 (x + 5)}}^{3}$ as $9 (x + 5)$ .

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Step 4.3.3.2.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{9 (x + 5)}$ as ${(9 (x + 5))}^{\frac{1}{3}}$ .

$f (\frac{\sqrt[3]{9 (x + 5)}}{3}) = 3 (\frac{{({(9 (x + 5))}^{\frac{1}{3}})}^{3}}{3^{3}}) - 5$

Step 4.3.3.2.1.2

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

$f (\frac{\sqrt[3]{9 (x + 5)}}{3}) = 3 (\frac{{(9 (x + 5))}^{\frac{1}{3} \cdot 3}}{3^{3}}) - 5$

Step 4.3.3.2.1.3

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

$f (\frac{\sqrt[3]{9 (x + 5)}}{3}) = 3 (\frac{{(9 (x + 5))}^{\frac{3}{3}}}{3^{3}}) - 5$

Step 4.3.3.2.1.4

Cancel the common factor of $3$ .

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Step 4.3.3.2.1.4.1

Cancel the common factor.

$f (\frac{\sqrt[3]{9 (x + 5)}}{3}) = 3 (\frac{{(9 (x + 5))}^{\frac{3}{3}}}{3^{3}}) - 5$

Step 4.3.3.2.1.4.2

Rewrite the expression.

$f (\frac{\sqrt[3]{9 (x + 5)}}{3}) = 3 (\frac{9 (x + 5)}{3^{3}}) - 5$

Step 4.3.3.2.1.5

Simplify.

$f (\frac{\sqrt[3]{9 (x + 5)}}{3}) = 3 (\frac{9 (x + 5)}{3^{3}}) - 5$

Step 4.3.3.2.2

Apply the distributive property.

$f (\frac{\sqrt[3]{9 (x + 5)}}{3}) = 3 (\frac{9 x + 9 \cdot 5}{3^{3}}) - 5$

Step 4.3.3.2.3

Multiply $9$ by $5$ .

$f (\frac{\sqrt[3]{9 (x + 5)}}{3}) = 3 (\frac{9 x + 45}{3^{3}}) - 5$

Step 4.3.3.2.4

Factor $9$ out of $9 x + 45$ .

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Step 4.3.3.2.4.1

Factor $9$ out of $9 x$ .

$f (\frac{\sqrt[3]{9 (x + 5)}}{3}) = 3 (\frac{9 (x) + 45}{3^{3}}) - 5$

Step 4.3.3.2.4.2

Factor $9$ out of $45$ .

$f (\frac{\sqrt[3]{9 (x + 5)}}{3}) = 3 (\frac{9 x + 9 \cdot 5}{3^{3}}) - 5$

Step 4.3.3.2.4.3

Factor $9$ out of $9 x + 9 \cdot 5$ .

$f (\frac{\sqrt[3]{9 (x + 5)}}{3}) = 3 (\frac{9 (x + 5)}{3^{3}}) - 5$

Step 4.3.3.3

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

$f (\frac{\sqrt[3]{9 (x + 5)}}{3}) = 3 (\frac{9 (x + 5)}{27}) - 5$

Step 4.3.3.4

Cancel the common factor of $3$ .

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Step 4.3.3.4.1

Factor $3$ out of $27$ .

$f (\frac{\sqrt[3]{9 (x + 5)}}{3}) = 3 (\frac{9 (x + 5)}{3 (9)}) - 5$

Step 4.3.3.4.2

Cancel the common factor.

$f (\frac{\sqrt[3]{9 (x + 5)}}{3}) = 3 (\frac{9 (x + 5)}{3 \cdot 9}) - 5$

Step 4.3.3.4.3

Rewrite the expression.

$f (\frac{\sqrt[3]{9 (x + 5)}}{3}) = \frac{9 (x + 5)}{9} - 5$

Step 4.3.3.5

Cancel the common factor of $9$ .

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Step 4.3.3.5.1

Cancel the common factor.

$f (\frac{\sqrt[3]{9 (x + 5)}}{3}) = \frac{9 (x + 5)}{9} - 5$

Step 4.3.3.5.2

Divide $x + 5$ by $1$ .

$f (\frac{\sqrt[3]{9 (x + 5)}}{3}) = x + 5 - 5$

Step 4.3.4

Combine the opposite terms in $x + 5 - 5$ .

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Step 4.3.4.1

Subtract $5$ from $5$ .

$f (\frac{\sqrt[3]{9 (x + 5)}}{3}) = x + 0$

Step 4.3.4.2

Add $x$ and $0$ .

$f (\frac{\sqrt[3]{9 (x + 5)}}{3}) = x$

Step 4.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{\sqrt[3]{9 (x + 5)}}{3}$ is the inverse of $f (x) = 3 x^{3} - 5$ .

$f^{- 1} (x) = \frac{\sqrt[3]{9 (x + 5)}}{3}$