Precalculus Examples

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

Step 1

Write $f (x) = 6 x^{3} - 3$ as an equation.

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

Step 2

Interchange the variables.

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

Step 3

Solve for $y$ .

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Step 3.1

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

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

Step 3.2

Add $3$ to both sides of the equation.

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

Step 3.3

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

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Step 3.3.1

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

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

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Cancel the common factor of $6$ .

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Step 3.3.2.1.1

Cancel the common factor.

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

Step 3.3.2.1.2

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

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

Step 3.3.3

Simplify the right side.

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Step 3.3.3.1

Cancel the common factor of $3$ and $6$ .

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Step 3.3.3.1.1

Factor $3$ out of $3$ .

$y^{3} = \frac{x}{6} + \frac{3 (1)}{6}$

Step 3.3.3.1.2

Cancel the common factors.

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Step 3.3.3.1.2.1

Factor $3$ out of $6$ .

$y^{3} = \frac{x}{6} + \frac{3 \cdot 1}{3 \cdot 2}$

Step 3.3.3.1.2.2

Cancel the common factor.

$y^{3} = \frac{x}{6} + \frac{3 \cdot 1}{3 \cdot 2}$

Step 3.3.3.1.2.3

Rewrite the expression.

$y^{3} = \frac{x}{6} + \frac{1}{2}$

Step 3.4

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

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

Step 3.5

Simplify $\sqrt[3]{\frac{x}{6} + \frac{1}{2}}$ .

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Step 3.5.1

To write $\frac{1}{2}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

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

Step 3.5.2

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.5.2.1

Multiply $\frac{1}{2}$ by $\frac{3}{3}$ .

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

Step 3.5.2.2

Multiply $2$ by $3$ .

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

Step 3.5.3

Combine the numerators over the common denominator.

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

Step 3.5.4

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

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

Step 3.5.5

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

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

Step 3.5.6

Combine and simplify the denominator.

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Step 3.5.6.1

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

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

Step 3.5.6.2

Raise $\sqrt[3]{6}$ to the power of $1$ .

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

Step 3.5.6.3

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

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

Step 3.5.6.4

Add $1$ and $2$ .

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

Step 3.5.6.5

Rewrite ${\sqrt[3]{6}}^{3}$ as $6$ .

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Step 3.5.6.5.1

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

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

Step 3.5.6.5.2

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

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

Step 3.5.6.5.3

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

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

Step 3.5.6.5.4

Cancel the common factor of $3$ .

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Step 3.5.6.5.4.1

Cancel the common factor.

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

Step 3.5.6.5.4.2

Rewrite the expression.

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

Step 3.5.6.5.5

Evaluate the exponent.

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

Step 3.5.7

Simplify the numerator.

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Step 3.5.7.1

Rewrite ${\sqrt[3]{6}}^{2}$ as $\sqrt[3]{6^{2}}$ .

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

Step 3.5.7.2

Raise $6$ to the power of $2$ .

$y = \frac{\sqrt[3]{x + 3} \sqrt[3]{36}}{6}$

Step 3.5.8

Simplify with factoring out.

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Step 3.5.8.1

Combine using the product rule for radicals.

$y = \frac{\sqrt[3]{(x + 3) \cdot 36}}{6}$

Step 3.5.8.2

Reorder factors in $\frac{\sqrt[3]{(x + 3) \cdot 36}}{6}$ .

$y = \frac{\sqrt[3]{36 (x + 3)}}{6}$

Step 4

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

$f^{- 1} (x) = \frac{\sqrt[3]{36 (x + 3)}}{6}$

Step 5

Verify if $f^{- 1} (x) = \frac{\sqrt[3]{36 (x + 3)}}{6}$ is the inverse of $f (x) = 6 x^{3} - 3$ .

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

$f^{- 1} (6 x^{3} - 3) = \frac{\sqrt[3]{36 ((6 x^{3} - 3) + 3)}}{6}$

Step 5.2.3

Simplify the numerator.

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Step 5.2.3.1

Add $- 3$ and $3$ .

$f^{- 1} (6 x^{3} - 3) = \frac{\sqrt[3]{36 (6 x^{3} + 0)}}{6}$

Step 5.2.3.2

Add $6 x^{3}$ and $0$ .

$f^{- 1} (6 x^{3} - 3) = \frac{\sqrt[3]{36 \cdot (6 x^{3})}}{6}$

Step 5.2.3.3

Multiply $36$ by $6$ .

$f^{- 1} (6 x^{3} - 3) = \frac{\sqrt[3]{216 x^{3}}}{6}$

Step 5.2.3.4

Rewrite $216 x^{3}$ as ${(6 x)}^{3}$ .

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

Step 5.2.3.5

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

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

Step 5.2.4

Cancel the common factor of $6$ .

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Step 5.2.4.1

Cancel the common factor.

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

Step 5.2.4.2

Divide $x$ by $1$ .

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

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 $f (\frac{\sqrt[3]{36 (x + 3)}}{6})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\frac{\sqrt[3]{36 (x + 3)}}{6}) = 6 {(\frac{\sqrt[3]{36 (x + 3)}}{6})}^{3} - 3$

Step 5.3.3

Simplify each term.

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Step 5.3.3.1

Apply the product rule to $\frac{\sqrt[3]{36 (x + 3)}}{6}$ .

$f (\frac{\sqrt[3]{36 (x + 3)}}{6}) = 6 (\frac{{\sqrt[3]{36 (x + 3)}}^{3}}{6^{3}}) - 3$

Step 5.3.3.2

Simplify the numerator.

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Step 5.3.3.2.1

Rewrite ${\sqrt[3]{36 (x + 3)}}^{3}$ as $36 (x + 3)$ .

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Step 5.3.3.2.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{36 (x + 3)}$ as ${(36 (x + 3))}^{\frac{1}{3}}$ .

$f (\frac{\sqrt[3]{36 (x + 3)}}{6}) = 6 (\frac{{({(36 (x + 3))}^{\frac{1}{3}})}^{3}}{6^{3}}) - 3$

Step 5.3.3.2.1.2

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

$f (\frac{\sqrt[3]{36 (x + 3)}}{6}) = 6 (\frac{{(36 (x + 3))}^{\frac{1}{3} \cdot 3}}{6^{3}}) - 3$

Step 5.3.3.2.1.3

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

$f (\frac{\sqrt[3]{36 (x + 3)}}{6}) = 6 (\frac{{(36 (x + 3))}^{\frac{3}{3}}}{6^{3}}) - 3$

Step 5.3.3.2.1.4

Cancel the common factor of $3$ .

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Step 5.3.3.2.1.4.1

Cancel the common factor.

$f (\frac{\sqrt[3]{36 (x + 3)}}{6}) = 6 (\frac{{(36 (x + 3))}^{\frac{3}{3}}}{6^{3}}) - 3$

Step 5.3.3.2.1.4.2

Rewrite the expression.

$f (\frac{\sqrt[3]{36 (x + 3)}}{6}) = 6 (\frac{36 (x + 3)}{6^{3}}) - 3$

Step 5.3.3.2.1.5

Simplify.

$f (\frac{\sqrt[3]{36 (x + 3)}}{6}) = 6 (\frac{36 (x + 3)}{6^{3}}) - 3$

Step 5.3.3.2.2

Apply the distributive property.

$f (\frac{\sqrt[3]{36 (x + 3)}}{6}) = 6 (\frac{36 x + 36 \cdot 3}{6^{3}}) - 3$

Step 5.3.3.2.3

Multiply $36$ by $3$ .

$f (\frac{\sqrt[3]{36 (x + 3)}}{6}) = 6 (\frac{36 x + 108}{6^{3}}) - 3$

Step 5.3.3.2.4

Factor $36$ out of $36 x + 108$ .

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Step 5.3.3.2.4.1

Factor $36$ out of $36 x$ .

$f (\frac{\sqrt[3]{36 (x + 3)}}{6}) = 6 (\frac{36 (x) + 108}{6^{3}}) - 3$

Step 5.3.3.2.4.2

Factor $36$ out of $108$ .

$f (\frac{\sqrt[3]{36 (x + 3)}}{6}) = 6 (\frac{36 x + 36 \cdot 3}{6^{3}}) - 3$

Step 5.3.3.2.4.3

Factor $36$ out of $36 x + 36 \cdot 3$ .

$f (\frac{\sqrt[3]{36 (x + 3)}}{6}) = 6 (\frac{36 (x + 3)}{6^{3}}) - 3$

Step 5.3.3.3

Raise $6$ to the power of $3$ .

$f (\frac{\sqrt[3]{36 (x + 3)}}{6}) = 6 (\frac{36 (x + 3)}{216}) - 3$

Step 5.3.3.4

Cancel the common factor of $6$ .

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Step 5.3.3.4.1

Factor $6$ out of $216$ .

$f (\frac{\sqrt[3]{36 (x + 3)}}{6}) = 6 (\frac{36 (x + 3)}{6 (36)}) - 3$

Step 5.3.3.4.2

Cancel the common factor.

$f (\frac{\sqrt[3]{36 (x + 3)}}{6}) = 6 (\frac{36 (x + 3)}{6 \cdot 36}) - 3$

Step 5.3.3.4.3

Rewrite the expression.

$f (\frac{\sqrt[3]{36 (x + 3)}}{6}) = \frac{36 (x + 3)}{36} - 3$

Step 5.3.3.5

Cancel the common factor of $36$ .

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Step 5.3.3.5.1

Cancel the common factor.

$f (\frac{\sqrt[3]{36 (x + 3)}}{6}) = \frac{36 (x + 3)}{36} - 3$

Step 5.3.3.5.2

Divide $x + 3$ by $1$ .

$f (\frac{\sqrt[3]{36 (x + 3)}}{6}) = x + 3 - 3$

Step 5.3.4

Combine the opposite terms in $x + 3 - 3$ .

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Step 5.3.4.1

Subtract $3$ from $3$ .

$f (\frac{\sqrt[3]{36 (x + 3)}}{6}) = x + 0$

Step 5.3.4.2

Add $x$ and $0$ .

$f (\frac{\sqrt[3]{36 (x + 3)}}{6}) = x$

Step 5.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{\sqrt[3]{36 (x + 3)}}{6}$ is the inverse of $f (x) = 6 x^{3} - 3$ .

$f^{- 1} (x) = \frac{\sqrt[3]{36 (x + 3)}}{6}$