Precalculus Examples

$f (x) = \frac{\sqrt[3]{x}}{2} + 9$ ; find $f^{- 1} (x)$

Step 1

Write $f (x) = \frac{\sqrt[3]{x}}{2} + 9$ as an equation.

$y = \frac{\sqrt[3]{x}}{2} + 9$

Step 2

Interchange the variables.

$x = \frac{\sqrt[3]{y}}{2} + 9$

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $\frac{\sqrt[3]{y}}{2} + 9 = x$ .

$\frac{\sqrt[3]{y}}{2} + 9 = x$

Step 3.2

Solve for $\sqrt[3]{y}$ .

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Step 3.2.1

Subtract $9$ from both sides of the equation.

$\frac{\sqrt[3]{y}}{2} = x - 9$

Step 3.2.2

Multiply both sides of the equation by $2$ .

$2 \frac{\sqrt[3]{y}}{2} = 2 (x - 9)$

Step 3.2.3

Simplify both sides of the equation.

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Step 3.2.3.1

Simplify the left side.

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Step 3.2.3.1.1

Cancel the common factor of $2$ .

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Step 3.2.3.1.1.1

Cancel the common factor.

$2 \frac{\sqrt[3]{y}}{2} = 2 (x - 9)$

Step 3.2.3.1.1.2

Rewrite the expression.

$\sqrt[3]{y} = 2 (x - 9)$

Step 3.2.3.2

Simplify the right side.

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Step 3.2.3.2.1

Simplify $2 (x - 9)$ .

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Step 3.2.3.2.1.1

Apply the distributive property.

$\sqrt[3]{y} = 2 x + 2 \cdot - 9$

Step 3.2.3.2.1.2

Multiply $2$ by $- 9$ .

$\sqrt[3]{y} = 2 x - 18$

Step 3.3

To remove the radical on the left side of the equation, cube both sides of the equation.

${\sqrt[3]{y}}^{3} = {(2 x - 18)}^{3}$

Step 3.4

Simplify each side of the equation.

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Step 3.4.1

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

${(y^{\frac{1}{3}})}^{3} = {(2 x - 18)}^{3}$

Step 3.4.2

Simplify the left side.

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Step 3.4.2.1

Simplify ${(y^{\frac{1}{3}})}^{3}$ .

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Step 3.4.2.1.1

Multiply the exponents in ${(y^{\frac{1}{3}})}^{3}$ .

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Step 3.4.2.1.1.1

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

$y^{\frac{1}{3} \cdot 3} = {(2 x - 18)}^{3}$

Step 3.4.2.1.1.2

Cancel the common factor of $3$ .

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Step 3.4.2.1.1.2.1

Cancel the common factor.

$y^{\frac{1}{3} \cdot 3} = {(2 x - 18)}^{3}$

Step 3.4.2.1.1.2.2

Rewrite the expression.

$y^{1} = {(2 x - 18)}^{3}$

Step 3.4.2.1.2

Simplify.

$y = {(2 x - 18)}^{3}$

Step 3.4.3

Simplify the right side.

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Step 3.4.3.1

Simplify ${(2 x - 18)}^{3}$ .

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Step 3.4.3.1.1

Use the Binomial Theorem.

$y = {(2 x)}^{3} + 3 {(2 x)}^{2} \cdot - 18 + 3 (2 x) {(- 18)}^{2} + {(- 18)}^{3}$

Step 3.4.3.1.2

Simplify each term.

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Step 3.4.3.1.2.1

Apply the product rule to $2 x$ .

$y = 2^{3} x^{3} + 3 {(2 x)}^{2} \cdot - 18 + 3 (2 x) {(- 18)}^{2} + {(- 18)}^{3}$

Step 3.4.3.1.2.2

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

$y = 8 x^{3} + 3 {(2 x)}^{2} \cdot - 18 + 3 (2 x) {(- 18)}^{2} + {(- 18)}^{3}$

Step 3.4.3.1.2.3

Apply the product rule to $2 x$ .

$y = 8 x^{3} + 3 (2^{2} x^{2}) \cdot - 18 + 3 (2 x) {(- 18)}^{2} + {(- 18)}^{3}$

Step 3.4.3.1.2.4

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

$y = 8 x^{3} + 3 (4 x^{2}) \cdot - 18 + 3 (2 x) {(- 18)}^{2} + {(- 18)}^{3}$

Step 3.4.3.1.2.5

Multiply $4$ by $3$ .

$y = 8 x^{3} + 12 x^{2} \cdot - 18 + 3 (2 x) {(- 18)}^{2} + {(- 18)}^{3}$

Step 3.4.3.1.2.6

Multiply $- 18$ by $12$ .

$y = 8 x^{3} - 216 x^{2} + 3 (2 x) {(- 18)}^{2} + {(- 18)}^{3}$

Step 3.4.3.1.2.7

Multiply $2$ by $3$ .

$y = 8 x^{3} - 216 x^{2} + 6 x {(- 18)}^{2} + {(- 18)}^{3}$

Step 3.4.3.1.2.8

Raise $- 18$ to the power of $2$ .

$y = 8 x^{3} - 216 x^{2} + 6 x \cdot 324 + {(- 18)}^{3}$

Step 3.4.3.1.2.9

Multiply $324$ by $6$ .

$y = 8 x^{3} - 216 x^{2} + 1944 x + {(- 18)}^{3}$

Step 3.4.3.1.2.10

Raise $- 18$ to the power of $3$ .

$y = 8 x^{3} - 216 x^{2} + 1944 x - 5832$

Step 4

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

$f^{- 1} (x) = 8 x^{3} - 216 x^{2} + 1944 x - 5832$

Step 5

Verify if $f^{- 1} (x) = 8 x^{3} - 216 x^{2} + 1944 x - 5832$ is the inverse of $f (x) = \frac{\sqrt[3]{x}}{2} + 9$ .

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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 $f^{- 1} (\frac{\sqrt[3]{x}}{2} + 9)$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\frac{\sqrt[3]{x}}{2} + 9) = 8 {(\frac{\sqrt[3]{x}}{2} + 9)}^{3} - 216 {(\frac{\sqrt[3]{x}}{2} + 9)}^{2} + 1944 (\frac{\sqrt[3]{x}}{2} + 9) - 5832$

Step 5.2.3

Simplify each term.

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Step 5.2.3.1

Use the Binomial Theorem.

$f^{- 1} (\frac{\sqrt[3]{x}}{2} + 9) = 8 ({(\frac{\sqrt[3]{x}}{2})}^{3} + 3 {(\frac{\sqrt[3]{x}}{2})}^{2} \cdot 9 + 3 (\frac{\sqrt[3]{x}}{2}) \cdot 9^{2} + 9^{3}) - 216 {(\frac{\sqrt[3]{x}}{2} + 9)}^{2} + 1944 (\frac{\sqrt[3]{x}}{2} + 9) - 5832$

Step 5.2.3.2

Simplify each term.

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Step 5.2.3.2.1

Apply the product rule to $\frac{\sqrt[3]{x}}{2}$ .

$f^{- 1} (\frac{\sqrt[3]{x}}{2} + 9) = 8 (\frac{{\sqrt[3]{x}}^{3}}{2^{3}} + 3 {(\frac{\sqrt[3]{x}}{2})}^{2} \cdot 9 + 3 (\frac{\sqrt[3]{x}}{2}) \cdot 9^{2} + 9^{3}) - 216 {(\frac{\sqrt[3]{x}}{2} + 9)}^{2} + 1944 (\frac{\sqrt[3]{x}}{2} + 9) - 5832$

Step 5.2.3.2.2

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

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Step 5.2.3.2.2.1

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

$f^{- 1} (\frac{\sqrt[3]{x}}{2} + 9) = 8 (\frac{{(x^{\frac{1}{3}})}^{3}}{2^{3}} + 3 {(\frac{\sqrt[3]{x}}{2})}^{2} \cdot 9 + 3 (\frac{\sqrt[3]{x}}{2}) \cdot 9^{2} + 9^{3}) - 216 {(\frac{\sqrt[3]{x}}{2} + 9)}^{2} + 1944 (\frac{\sqrt[3]{x}}{2} + 9) - 5832$

Step 5.2.3.2.2.2

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

$f^{- 1} (\frac{\sqrt[3]{x}}{2} + 9) = 8 (\frac{x^{\frac{1}{3} \cdot 3}}{2^{3}} + 3 {(\frac{\sqrt[3]{x}}{2})}^{2} \cdot 9 + 3 (\frac{\sqrt[3]{x}}{2}) \cdot 9^{2} + 9^{3}) - 216 {(\frac{\sqrt[3]{x}}{2} + 9)}^{2} + 1944 (\frac{\sqrt[3]{x}}{2} + 9) - 5832$

Step 5.2.3.2.2.3

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

$f^{- 1} (\frac{\sqrt[3]{x}}{2} + 9) = 8 (\frac{x^{\frac{3}{3}}}{2^{3}} + 3 {(\frac{\sqrt[3]{x}}{2})}^{2} \cdot 9 + 3 (\frac{\sqrt[3]{x}}{2}) \cdot 9^{2} + 9^{3}) - 216 {(\frac{\sqrt[3]{x}}{2} + 9)}^{2} + 1944 (\frac{\sqrt[3]{x}}{2} + 9) - 5832$

Step 5.2.3.2.2.4

Cancel the common factor of $3$ .

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Step 5.2.3.2.2.4.1

Cancel the common factor.

Step 5.2.3.2.2.4.2

Rewrite the expression.

$f^{- 1} (\frac{\sqrt[3]{x}}{2} + 9) = 8 (\frac{x}{2^{3}} + 3 {(\frac{\sqrt[3]{x}}{2})}^{2} \cdot 9 + 3 (\frac{\sqrt[3]{x}}{2}) \cdot 9^{2} + 9^{3}) - 216 {(\frac{\sqrt[3]{x}}{2} + 9)}^{2} + 1944 (\frac{\sqrt[3]{x}}{2} + 9) - 5832$

Step 5.2.3.2.2.5

Simplify.

Step 5.2.3.2.3

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

$f^{- 1} (\frac{\sqrt[3]{x}}{2} + 9) = 8 (\frac{x}{8} + 3 {(\frac{\sqrt[3]{x}}{2})}^{2} \cdot 9 + 3 (\frac{\sqrt[3]{x}}{2}) \cdot 9^{2} + 9^{3}) - 216 {(\frac{\sqrt[3]{x}}{2} + 9)}^{2} + 1944 (\frac{\sqrt[3]{x}}{2} + 9) - 5832$

Step 5.2.3.2.4

Apply the product rule to $\frac{\sqrt[3]{x}}{2}$ .

$f^{- 1} (\frac{\sqrt[3]{x}}{2} + 9) = 8 (\frac{x}{8} + 3 (\frac{{\sqrt[3]{x}}^{2}}{2^{2}}) \cdot 9 + 3 (\frac{\sqrt[3]{x}}{2}) \cdot 9^{2} + 9^{3}) - 216 {(\frac{\sqrt[3]{x}}{2} + 9)}^{2} + 1944 (\frac{\sqrt[3]{x}}{2} + 9) - 5832$

Step 5.2.3.2.5

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

$f^{- 1} (\frac{\sqrt[3]{x}}{2} + 9) = 8 (\frac{x}{8} + 3 (\frac{\sqrt[3]{x^{2}}}{2^{2}}) \cdot 9 + 3 (\frac{\sqrt[3]{x}}{2}) \cdot 9^{2} + 9^{3}) - 216 {(\frac{\sqrt[3]{x}}{2} + 9)}^{2} + 1944 (\frac{\sqrt[3]{x}}{2} + 9) - 5832$

Step 5.2.3.2.6

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

$f^{- 1} (\frac{\sqrt[3]{x}}{2} + 9) = 8 (\frac{x}{8} + 3 (\frac{\sqrt[3]{x^{2}}}{4}) \cdot 9 + 3 (\frac{\sqrt[3]{x}}{2}) \cdot 9^{2} + 9^{3}) - 216 {(\frac{\sqrt[3]{x}}{2} + 9)}^{2} + 1944 (\frac{\sqrt[3]{x}}{2} + 9) - 5832$

Step 5.2.3.2.7

Combine $3$ and $\frac{\sqrt[3]{x^{2}}}{4}$ .

$f^{- 1} (\frac{\sqrt[3]{x}}{2} + 9) = 8 (\frac{x}{8} + \frac{3 \sqrt[3]{x^{2}}}{4} \cdot 9 + 3 (\frac{\sqrt[3]{x}}{2}) \cdot 9^{2} + 9^{3}) - 216 {(\frac{\sqrt[3]{x}}{2} + 9)}^{2} + 1944 (\frac{\sqrt[3]{x}}{2} + 9) - 5832$

Step 5.2.3.2.8

Multiply $\frac{3 \sqrt[3]{x^{2}}}{4} \cdot 9$ .

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Step 5.2.3.2.8.1

Combine $\frac{3 \sqrt[3]{x^{2}}}{4}$ and $9$ .

$f^{- 1} (\frac{\sqrt[3]{x}}{2} + 9) = 8 (\frac{x}{8} + \frac{3 \sqrt[3]{x^{2}} \cdot 9}{4} + 3 (\frac{\sqrt[3]{x}}{2}) \cdot 9^{2} + 9^{3}) - 216 {(\frac{\sqrt[3]{x}}{2} + 9)}^{2} + 1944 (\frac{\sqrt[3]{x}}{2} + 9) - 5832$

Step 5.2.3.2.8.2

Multiply $9$ by $3$ .

$f^{- 1} (\frac{\sqrt[3]{x}}{2} + 9) = 8 (\frac{x}{8} + \frac{27 \sqrt[3]{x^{2}}}{4} + 3 (\frac{\sqrt[3]{x}}{2}) \cdot 9^{2} + 9^{3}) - 216 {(\frac{\sqrt[3]{x}}{2} + 9)}^{2} + 1944 (\frac{\sqrt[3]{x}}{2} + 9) - 5832$

Step 5.2.3.2.9

Combine $3$ and $\frac{\sqrt[3]{x}}{2}$ .

$f^{- 1} (\frac{\sqrt[3]{x}}{2} + 9) = 8 (\frac{x}{8} + \frac{27 \sqrt[3]{x^{2}}}{4} + \frac{3 \sqrt[3]{x}}{2} \cdot 9^{2} + 9^{3}) - 216 {(\frac{\sqrt[3]{x}}{2} + 9)}^{2} + 1944 (\frac{\sqrt[3]{x}}{2} + 9) - 5832$

Step 5.2.3.2.10

Raise $9$ to the power of $2$ .

$f^{- 1} (\frac{\sqrt[3]{x}}{2} + 9) = 8 (\frac{x}{8} + \frac{27 \sqrt[3]{x^{2}}}{4} + \frac{3 \sqrt[3]{x}}{2} \cdot 81 + 9^{3}) - 216 {(\frac{\sqrt[3]{x}}{2} + 9)}^{2} + 1944 (\frac{\sqrt[3]{x}}{2} + 9) - 5832$

Step 5.2.3.2.11

Multiply $\frac{3 \sqrt[3]{x}}{2} \cdot 81$ .

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Step 5.2.3.2.11.1

Combine $\frac{3 \sqrt[3]{x}}{2}$ and $81$ .

$f^{- 1} (\frac{\sqrt[3]{x}}{2} + 9) = 8 (\frac{x}{8} + \frac{27 \sqrt[3]{x^{2}}}{4} + \frac{3 \sqrt[3]{x} \cdot 81}{2} + 9^{3}) - 216 {(\frac{\sqrt[3]{x}}{2} + 9)}^{2} + 1944 (\frac{\sqrt[3]{x}}{2} + 9) - 5832$

Step 5.2.3.2.11.2

Multiply $81$ by $3$ .

$f^{- 1} (\frac{\sqrt[3]{x}}{2} + 9) = 8 (\frac{x}{8} + \frac{27 \sqrt[3]{x^{2}}}{4} + \frac{243 \sqrt[3]{x}}{2} + 9^{3}) - 216 {(\frac{\sqrt[3]{x}}{2} + 9)}^{2} + 1944 (\frac{\sqrt[3]{x}}{2} + 9) - 5832$

Step 5.2.3.2.12

Raise $9$ to the power of $3$ .

$f^{- 1} (\frac{\sqrt[3]{x}}{2} + 9) = 8 (\frac{x}{8} + \frac{27 \sqrt[3]{x^{2}}}{4} + \frac{243 \sqrt[3]{x}}{2} + 729) - 216 {(\frac{\sqrt[3]{x}}{2} + 9)}^{2} + 1944 (\frac{\sqrt[3]{x}}{2} + 9) - 5832$

Step 5.2.3.3

Apply the distributive property.

$f^{- 1} (\frac{\sqrt[3]{x}}{2} + 9) = 8 (\frac{x}{8}) + 8 (\frac{27 \sqrt[3]{x^{2}}}{4}) + 8 (\frac{243 \sqrt[3]{x}}{2}) + 8 \cdot 729 - 216 {(\frac{\sqrt[3]{x}}{2} + 9)}^{2} + 1944 (\frac{\sqrt[3]{x}}{2} + 9) - 5832$

Step 5.2.3.4

Simplify.

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Step 5.2.3.4.1

Cancel the common factor of $8$ .

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Step 5.2.3.4.1.1

Cancel the common factor.

Step 5.2.3.4.1.2

Rewrite the expression.

$f^{- 1} (\frac{\sqrt[3]{x}}{2} + 9) = x + 8 (\frac{27 \sqrt[3]{x^{2}}}{4}) + 8 (\frac{243 \sqrt[3]{x}}{2}) + 8 \cdot 729 - 216 {(\frac{\sqrt[3]{x}}{2} + 9)}^{2} + 1944 (\frac{\sqrt[3]{x}}{2} + 9) - 5832$

Step 5.2.3.4.2

Cancel the common factor of $4$ .

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Step 5.2.3.4.2.1

Factor $4$ out of $8$ .

$f^{- 1} (\frac{\sqrt[3]{x}}{2} + 9) = x + 4 (2) (\frac{27 \sqrt[3]{x^{2}}}{4}) + 8 (\frac{243 \sqrt[3]{x}}{2}) + 8 \cdot 729 - 216 {(\frac{\sqrt[3]{x}}{2} + 9)}^{2} + 1944 (\frac{\sqrt[3]{x}}{2} + 9) - 5832$

Step 5.2.3.4.2.2

Cancel the common factor.

$f^{- 1} (\frac{\sqrt[3]{x}}{2} + 9) = x + 4 \cdot (2 (\frac{27 \sqrt[3]{x^{2}}}{4})) + 8 (\frac{243 \sqrt[3]{x}}{2}) + 8 \cdot 729 - 216 {(\frac{\sqrt[3]{x}}{2} + 9)}^{2} + 1944 (\frac{\sqrt[3]{x}}{2} + 9) - 5832$

Step 5.2.3.4.2.3

Rewrite the expression.

$f^{- 1} (\frac{\sqrt[3]{x}}{2} + 9) = x + 2 (27 \sqrt[3]{x^{2}}) + 8 (\frac{243 \sqrt[3]{x}}{2}) + 8 \cdot 729 - 216 {(\frac{\sqrt[3]{x}}{2} + 9)}^{2} + 1944 (\frac{\sqrt[3]{x}}{2} + 9) - 5832$

Step 5.2.3.4.3

Multiply $27$ by $2$ .

$f^{- 1} (\frac{\sqrt[3]{x}}{2} + 9) = x + 54 \sqrt[3]{x^{2}} + 8 (\frac{243 \sqrt[3]{x}}{2}) + 8 \cdot 729 - 216 {(\frac{\sqrt[3]{x}}{2} + 9)}^{2} + 1944 (\frac{\sqrt[3]{x}}{2} + 9) - 5832$

Step 5.2.3.4.4

Cancel the common factor of $2$ .

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Step 5.2.3.4.4.1

Factor $2$ out of $8$ .

$f^{- 1} (\frac{\sqrt[3]{x}}{2} + 9) = x + 54 \sqrt[3]{x^{2}} + 2 (4) (\frac{243 \sqrt[3]{x}}{2}) + 8 \cdot 729 - 216 {(\frac{\sqrt[3]{x}}{2} + 9)}^{2} + 1944 (\frac{\sqrt[3]{x}}{2} + 9) - 5832$

Step 5.2.3.4.4.2

Cancel the common factor.

$f^{- 1} (\frac{\sqrt[3]{x}}{2} + 9) = x + 54 \sqrt[3]{x^{2}} + 2 \cdot (4 (\frac{243 \sqrt[3]{x}}{2})) + 8 \cdot 729 - 216 {(\frac{\sqrt[3]{x}}{2} + 9)}^{2} + 1944 (\frac{\sqrt[3]{x}}{2} + 9) - 5832$

Step 5.2.3.4.4.3

Rewrite the expression.

$f^{- 1} (\frac{\sqrt[3]{x}}{2} + 9) = x + 54 \sqrt[3]{x^{2}} + 4 (243 \sqrt[3]{x}) + 8 \cdot 729 - 216 {(\frac{\sqrt[3]{x}}{2} + 9)}^{2} + 1944 (\frac{\sqrt[3]{x}}{2} + 9) - 5832$

Step 5.2.3.4.5

Multiply $243$ by $4$ .

$f^{- 1} (\frac{\sqrt[3]{x}}{2} + 9) = x + 54 \sqrt[3]{x^{2}} + 972 \sqrt[3]{x} + 8 \cdot 729 - 216 {(\frac{\sqrt[3]{x}}{2} + 9)}^{2} + 1944 (\frac{\sqrt[3]{x}}{2} + 9) - 5832$

Step 5.2.3.4.6

Multiply $8$ by $729$ .

$f^{- 1} (\frac{\sqrt[3]{x}}{2} + 9) = x + 54 \sqrt[3]{x^{2}} + 972 \sqrt[3]{x} + 5832 - 216 {(\frac{\sqrt[3]{x}}{2} + 9)}^{2} + 1944 (\frac{\sqrt[3]{x}}{2} + 9) - 5832$

Step 5.2.3.5

Rewrite ${(\frac{\sqrt[3]{x}}{2} + 9)}^{2}$ as $(\frac{\sqrt[3]{x}}{2} + 9) (\frac{\sqrt[3]{x}}{2} + 9)$ .

$f^{- 1} (\frac{\sqrt[3]{x}}{2} + 9) = x + 54 \sqrt[3]{x^{2}} + 972 \sqrt[3]{x} + 5832 - 216 ((\frac{\sqrt[3]{x}}{2} + 9) (\frac{\sqrt[3]{x}}{2} + 9)) + 1944 (\frac{\sqrt[3]{x}}{2} + 9) - 5832$

Step 5.2.3.6

Expand $(\frac{\sqrt[3]{x}}{2} + 9) (\frac{\sqrt[3]{x}}{2} + 9)$ using the FOIL Method.

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Step 5.2.3.6.1

Apply the distributive property.

$f^{- 1} (\frac{\sqrt[3]{x}}{2} + 9) = x + 54 \sqrt[3]{x^{2}} + 972 \sqrt[3]{x} + 5832 - 216 (\frac{\sqrt[3]{x}}{2} \cdot (\frac{\sqrt[3]{x}}{2} + 9) + 9 (\frac{\sqrt[3]{x}}{2} + 9)) + 1944 (\frac{\sqrt[3]{x}}{2} + 9) - 5832$

Step 5.2.3.6.2

Apply the distributive property.

$f^{- 1} (\frac{\sqrt[3]{x}}{2} + 9) = x + 54 \sqrt[3]{x^{2}} + 972 \sqrt[3]{x} + 5832 - 216 (\frac{\sqrt[3]{x}}{2} \cdot \frac{\sqrt[3]{x}}{2} + \frac{\sqrt[3]{x}}{2} \cdot 9 + 9 (\frac{\sqrt[3]{x}}{2} + 9)) + 1944 (\frac{\sqrt[3]{x}}{2} + 9) - 5832$

Step 5.2.3.6.3

Apply the distributive property.

Step 5.2.3.7

Simplify and combine like terms.

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Step 5.2.3.7.1

Simplify each term.

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Step 5.2.3.7.1.1

Multiply $\frac{\sqrt[3]{x}}{2} \cdot \frac{\sqrt[3]{x}}{2}$ .

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Step 5.2.3.7.1.1.1

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

$f^{- 1} (\frac{\sqrt[3]{x}}{2} + 9) = x + 54 \sqrt[3]{x^{2}} + 972 \sqrt[3]{x} + 5832 - 216 (\frac{\sqrt[3]{x} \sqrt[3]{x}}{2 \cdot 2} + \frac{\sqrt[3]{x}}{2} \cdot 9 + 9 (\frac{\sqrt[3]{x}}{2}) + 9 \cdot 9) + 1944 (\frac{\sqrt[3]{x}}{2} + 9) - 5832$

Step 5.2.3.7.1.1.2

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

Step 5.2.3.7.1.1.3

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

Step 5.2.3.7.1.1.4

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

$f^{- 1} (\frac{\sqrt[3]{x}}{2} + 9) = x + 54 \sqrt[3]{x^{2}} + 972 \sqrt[3]{x} + 5832 - 216 (\frac{{\sqrt[3]{x}}^{1 + 1}}{2 \cdot 2} + \frac{\sqrt[3]{x}}{2} \cdot 9 + 9 (\frac{\sqrt[3]{x}}{2}) + 9 \cdot 9) + 1944 (\frac{\sqrt[3]{x}}{2} + 9) - 5832$

Step 5.2.3.7.1.1.5

Add $1$ and $1$ .

$f^{- 1} (\frac{\sqrt[3]{x}}{2} + 9) = x + 54 \sqrt[3]{x^{2}} + 972 \sqrt[3]{x} + 5832 - 216 (\frac{{\sqrt[3]{x}}^{2}}{2 \cdot 2} + \frac{\sqrt[3]{x}}{2} \cdot 9 + 9 (\frac{\sqrt[3]{x}}{2}) + 9 \cdot 9) + 1944 (\frac{\sqrt[3]{x}}{2} + 9) - 5832$

Step 5.2.3.7.1.1.6

Multiply $2$ by $2$ .

$f^{- 1} (\frac{\sqrt[3]{x}}{2} + 9) = x + 54 \sqrt[3]{x^{2}} + 972 \sqrt[3]{x} + 5832 - 216 (\frac{{\sqrt[3]{x}}^{2}}{4} + \frac{\sqrt[3]{x}}{2} \cdot 9 + 9 (\frac{\sqrt[3]{x}}{2}) + 9 \cdot 9) + 1944 (\frac{\sqrt[3]{x}}{2} + 9) - 5832$

Step 5.2.3.7.1.2

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

$f^{- 1} (\frac{\sqrt[3]{x}}{2} + 9) = x + 54 \sqrt[3]{x^{2}} + 972 \sqrt[3]{x} + 5832 - 216 (\frac{\sqrt[3]{x^{2}}}{4} + \frac{\sqrt[3]{x}}{2} \cdot 9 + 9 (\frac{\sqrt[3]{x}}{2}) + 9 \cdot 9) + 1944 (\frac{\sqrt[3]{x}}{2} + 9) - 5832$

Step 5.2.3.7.1.3

Combine $\frac{\sqrt[3]{x}}{2}$ and $9$ .

$f^{- 1} (\frac{\sqrt[3]{x}}{2} + 9) = x + 54 \sqrt[3]{x^{2}} + 972 \sqrt[3]{x} + 5832 - 216 (\frac{\sqrt[3]{x^{2}}}{4} + \frac{\sqrt[3]{x} \cdot 9}{2} + 9 (\frac{\sqrt[3]{x}}{2}) + 9 \cdot 9) + 1944 (\frac{\sqrt[3]{x}}{2} + 9) - 5832$

Step 5.2.3.7.1.4

Move $9$ to the left of $\sqrt[3]{x}$ .

$f^{- 1} (\frac{\sqrt[3]{x}}{2} + 9) = x + 54 \sqrt[3]{x^{2}} + 972 \sqrt[3]{x} + 5832 - 216 (\frac{\sqrt[3]{x^{2}}}{4} + \frac{9 \cdot \sqrt[3]{x}}{2} + 9 (\frac{\sqrt[3]{x}}{2}) + 9 \cdot 9) + 1944 (\frac{\sqrt[3]{x}}{2} + 9) - 5832$

Step 5.2.3.7.1.5

Combine $9$ and $\frac{\sqrt[3]{x}}{2}$ .

$f^{- 1} (\frac{\sqrt[3]{x}}{2} + 9) = x + 54 \sqrt[3]{x^{2}} + 972 \sqrt[3]{x} + 5832 - 216 (\frac{\sqrt[3]{x^{2}}}{4} + \frac{9 \sqrt[3]{x}}{2} + \frac{9 \sqrt[3]{x}}{2} + 9 \cdot 9) + 1944 (\frac{\sqrt[3]{x}}{2} + 9) - 5832$

Step 5.2.3.7.1.6

Multiply $9$ by $9$ .

$f^{- 1} (\frac{\sqrt[3]{x}}{2} + 9) = x + 54 \sqrt[3]{x^{2}} + 972 \sqrt[3]{x} + 5832 - 216 (\frac{\sqrt[3]{x^{2}}}{4} + \frac{9 \sqrt[3]{x}}{2} + \frac{9 \sqrt[3]{x}}{2} + 81) + 1944 (\frac{\sqrt[3]{x}}{2} + 9) - 5832$

Step 5.2.3.7.2

Add $\frac{9 \sqrt[3]{x}}{2}$ and $\frac{9 \sqrt[3]{x}}{2}$ .

$f^{- 1} (\frac{\sqrt[3]{x}}{2} + 9) = x + 54 \sqrt[3]{x^{2}} + 972 \sqrt[3]{x} + 5832 - 216 (\frac{\sqrt[3]{x^{2}}}{4} + 2 (\frac{9 \sqrt[3]{x}}{2}) + 81) + 1944 (\frac{\sqrt[3]{x}}{2} + 9) - 5832$

Step 5.2.3.8

Cancel the common factor of $2$ .

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Step 5.2.3.8.1

Cancel the common factor.

$f^{- 1} (\frac{\sqrt[3]{x}}{2} + 9) = x + 54 \sqrt[3]{x^{2}} + 972 \sqrt[3]{x} + 5832 - 216 (\frac{\sqrt[3]{x^{2}}}{4} + 2 (\frac{9 \sqrt[3]{x}}{2}) + 81) + 1944 (\frac{\sqrt[3]{x}}{2} + 9) - 5832$

Step 5.2.3.8.2

Rewrite the expression.

$f^{- 1} (\frac{\sqrt[3]{x}}{2} + 9) = x + 54 \sqrt[3]{x^{2}} + 972 \sqrt[3]{x} + 5832 - 216 (\frac{\sqrt[3]{x^{2}}}{4} + 9 \sqrt[3]{x} + 81) + 1944 (\frac{\sqrt[3]{x}}{2} + 9) - 5832$

Step 5.2.3.9

Apply the distributive property.

$f^{- 1} (\frac{\sqrt[3]{x}}{2} + 9) = x + 54 \sqrt[3]{x^{2}} + 972 \sqrt[3]{x} + 5832 - 216 \frac{\sqrt[3]{x^{2}}}{4} - 216 (9 \sqrt[3]{x}) - 216 \cdot 81 + 1944 (\frac{\sqrt[3]{x}}{2} + 9) - 5832$

Step 5.2.3.10

Simplify.

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Step 5.2.3.10.1

Cancel the common factor of $4$ .

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Step 5.2.3.10.1.1

Factor $4$ out of $- 216$ .

$f^{- 1} (\frac{\sqrt[3]{x}}{2} + 9) = x + 54 \sqrt[3]{x^{2}} + 972 \sqrt[3]{x} + 5832 + 4 (- 54) (\frac{\sqrt[3]{x^{2}}}{4}) - 216 (9 \sqrt[3]{x}) - 216 \cdot 81 + 1944 (\frac{\sqrt[3]{x}}{2} + 9) - 5832$

Step 5.2.3.10.1.2

Cancel the common factor.

$f^{- 1} (\frac{\sqrt[3]{x}}{2} + 9) = x + 54 \sqrt[3]{x^{2}} + 972 \sqrt[3]{x} + 5832 + 4 \cdot (- 54 \frac{\sqrt[3]{x^{2}}}{4}) - 216 (9 \sqrt[3]{x}) - 216 \cdot 81 + 1944 (\frac{\sqrt[3]{x}}{2} + 9) - 5832$

Step 5.2.3.10.1.3

Rewrite the expression.

$f^{- 1} (\frac{\sqrt[3]{x}}{2} + 9) = x + 54 \sqrt[3]{x^{2}} + 972 \sqrt[3]{x} + 5832 - 54 \sqrt[3]{x^{2}} - 216 (9 \sqrt[3]{x}) - 216 \cdot 81 + 1944 (\frac{\sqrt[3]{x}}{2} + 9) - 5832$

Step 5.2.3.10.2

Multiply $9$ by $- 216$ .

$f^{- 1} (\frac{\sqrt[3]{x}}{2} + 9) = x + 54 \sqrt[3]{x^{2}} + 972 \sqrt[3]{x} + 5832 - 54 \sqrt[3]{x^{2}} - 1944 \sqrt[3]{x} - 216 \cdot 81 + 1944 (\frac{\sqrt[3]{x}}{2} + 9) - 5832$

Step 5.2.3.10.3

Multiply $- 216$ by $81$ .

$f^{- 1} (\frac{\sqrt[3]{x}}{2} + 9) = x + 54 \sqrt[3]{x^{2}} + 972 \sqrt[3]{x} + 5832 - 54 \sqrt[3]{x^{2}} - 1944 \sqrt[3]{x} - 17496 + 1944 (\frac{\sqrt[3]{x}}{2} + 9) - 5832$

Step 5.2.3.11

Apply the distributive property.

$f^{- 1} (\frac{\sqrt[3]{x}}{2} + 9) = x + 54 \sqrt[3]{x^{2}} + 972 \sqrt[3]{x} + 5832 - 54 \sqrt[3]{x^{2}} - 1944 \sqrt[3]{x} - 17496 + 1944 (\frac{\sqrt[3]{x}}{2}) + 1944 \cdot 9 - 5832$

Step 5.2.3.12

Cancel the common factor of $2$ .

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Step 5.2.3.12.1

Factor $2$ out of $1944$ .

$f^{- 1} (\frac{\sqrt[3]{x}}{2} + 9) = x + 54 \sqrt[3]{x^{2}} + 972 \sqrt[3]{x} + 5832 - 54 \sqrt[3]{x^{2}} - 1944 \sqrt[3]{x} - 17496 + 2 (972) (\frac{\sqrt[3]{x}}{2}) + 1944 \cdot 9 - 5832$

Step 5.2.3.12.2

Cancel the common factor.

$f^{- 1} (\frac{\sqrt[3]{x}}{2} + 9) = x + 54 \sqrt[3]{x^{2}} + 972 \sqrt[3]{x} + 5832 - 54 \sqrt[3]{x^{2}} - 1944 \sqrt[3]{x} - 17496 + 2 \cdot (972 (\frac{\sqrt[3]{x}}{2})) + 1944 \cdot 9 - 5832$

Step 5.2.3.12.3

Rewrite the expression.

$f^{- 1} (\frac{\sqrt[3]{x}}{2} + 9) = x + 54 \sqrt[3]{x^{2}} + 972 \sqrt[3]{x} + 5832 - 54 \sqrt[3]{x^{2}} - 1944 \sqrt[3]{x} - 17496 + 972 \sqrt[3]{x} + 1944 \cdot 9 - 5832$

Step 5.2.3.13

Multiply $1944$ by $9$ .

$f^{- 1} (\frac{\sqrt[3]{x}}{2} + 9) = x + 54 \sqrt[3]{x^{2}} + 972 \sqrt[3]{x} + 5832 - 54 \sqrt[3]{x^{2}} - 1944 \sqrt[3]{x} - 17496 + 972 \sqrt[3]{x} + 17496 - 5832$

Step 5.2.4

Simplify by adding terms.

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Step 5.2.4.1

Combine the opposite terms in $x + 54 \sqrt[3]{x^{2}} + 972 \sqrt[3]{x} + 5832 - 54 \sqrt[3]{x^{2}} - 1944 \sqrt[3]{x} - 17496 + 972 \sqrt[3]{x} + 17496 - 5832$ .

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Step 5.2.4.1.1

Subtract $54 \sqrt[3]{x^{2}}$ from $54 \sqrt[3]{x^{2}}$ .

$f^{- 1} (\frac{\sqrt[3]{x}}{2} + 9) = x + 0 + 972 \sqrt[3]{x} + 5832 - 1944 \sqrt[3]{x} - 17496 + 972 \sqrt[3]{x} + 17496 - 5832$

Step 5.2.4.1.2

Add $x$ and $0$ .

$f^{- 1} (\frac{\sqrt[3]{x}}{2} + 9) = x + 972 \sqrt[3]{x} + 5832 - 1944 \sqrt[3]{x} - 17496 + 972 \sqrt[3]{x} + 17496 - 5832$

Step 5.2.4.1.3

Add $- 17496$ and $17496$ .

$f^{- 1} (\frac{\sqrt[3]{x}}{2} + 9) = x + 972 \sqrt[3]{x} + 5832 - 1944 \sqrt[3]{x} + 0 + 972 \sqrt[3]{x} - 5832$

Step 5.2.4.1.4

Add $x + 972 \sqrt[3]{x} + 5832 - 1944 \sqrt[3]{x}$ and $0$ .

$f^{- 1} (\frac{\sqrt[3]{x}}{2} + 9) = x + 972 \sqrt[3]{x} + 5832 - 1944 \sqrt[3]{x} + 972 \sqrt[3]{x} - 5832$

Step 5.2.4.1.5

Subtract $5832$ from $5832$ .

$f^{- 1} (\frac{\sqrt[3]{x}}{2} + 9) = x + 972 \sqrt[3]{x} + 0 - 1944 \sqrt[3]{x} + 972 \sqrt[3]{x}$

Step 5.2.4.1.6

Add $x + 972 \sqrt[3]{x}$ and $0$ .

$f^{- 1} (\frac{\sqrt[3]{x}}{2} + 9) = x + 972 \sqrt[3]{x} - 1944 \sqrt[3]{x} + 972 \sqrt[3]{x}$

Step 5.2.4.2

Subtract $1944 \sqrt[3]{x}$ from $972 \sqrt[3]{x}$ .

$f^{- 1} (\frac{\sqrt[3]{x}}{2} + 9) = x - 972 \sqrt[3]{x} + 972 \sqrt[3]{x}$

Step 5.2.4.3

Combine the opposite terms in $x - 972 \sqrt[3]{x} + 972 \sqrt[3]{x}$ .

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Step 5.2.4.3.1

Add $- 972 \sqrt[3]{x}$ and $972 \sqrt[3]{x}$ .

$f^{- 1} (\frac{\sqrt[3]{x}}{2} + 9) = x + 0$

Step 5.2.4.3.2

Add $x$ and $0$ .

$f^{- 1} (\frac{\sqrt[3]{x}}{2} + 9) = x$

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 $f (8 x^{3} - 216 x^{2} + 1944 x - 5832)$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (8 x^{3} - 216 x^{2} + 1944 x - 5832) = \frac{\sqrt[3]{8 x^{3} - 216 x^{2} + 1944 x - 5832}}{2} + 9$

Step 5.3.3

Simplify each term.

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Step 5.3.3.1

Simplify the numerator.

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Step 5.3.3.1.1

Factor $8$ out of $8 x^{3} - 216 x^{2} + 1944 x - 5832$ .

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Step 5.3.3.1.1.1

Factor $8$ out of $8 x^{3}$ .

$f (8 x^{3} - 216 x^{2} + 1944 x - 5832) = \frac{\sqrt[3]{8 (x^{3}) - 216 x^{2} + 1944 x - 5832}}{2} + 9$

Step 5.3.3.1.1.2

Factor $8$ out of $- 216 x^{2}$ .

$f (8 x^{3} - 216 x^{2} + 1944 x - 5832) = \frac{\sqrt[3]{8 (x^{3}) + 8 (- 27 x^{2}) + 1944 x - 5832}}{2} + 9$

Step 5.3.3.1.1.3

Factor $8$ out of $1944 x$ .

$f (8 x^{3} - 216 x^{2} + 1944 x - 5832) = \frac{\sqrt[3]{8 (x^{3}) + 8 (- 27 x^{2}) + 8 (243 x) - 5832}}{2} + 9$

Step 5.3.3.1.1.4

Factor $8$ out of $- 5832$ .

$f (8 x^{3} - 216 x^{2} + 1944 x - 5832) = \frac{\sqrt[3]{8 x^{3} + 8 (- 27 x^{2}) + 8 (243 x) + 8 \cdot - 729}}{2} + 9$

Step 5.3.3.1.1.5

Factor $8$ out of $8 x^{3} + 8 (- 27 x^{2})$ .

$f (8 x^{3} - 216 x^{2} + 1944 x - 5832) = \frac{\sqrt[3]{8 (x^{3} - 27 x^{2}) + 8 (243 x) + 8 \cdot - 729}}{2} + 9$

Step 5.3.3.1.1.6

Factor $8$ out of $8 (x^{3} - 27 x^{2}) + 8 (243 x)$ .

$f (8 x^{3} - 216 x^{2} + 1944 x - 5832) = \frac{\sqrt[3]{8 (x^{3} - 27 x^{2} + 243 x) + 8 \cdot - 729}}{2} + 9$

Step 5.3.3.1.1.7

Factor $8$ out of $8 (x^{3} - 27 x^{2} + 243 x) + 8 \cdot - 729$ .

$f (8 x^{3} - 216 x^{2} + 1944 x - 5832) = \frac{\sqrt[3]{8 (x^{3} - 27 x^{2} + 243 x - 729)}}{2} + 9$

Step 5.3.3.1.2

Rewrite $8 (x^{3} - 27 x^{2} + 243 x - 729)$ as $2^{3} (x^{3} + {(- 3)}^{3} x^{2} + 243 x - 729)$ .

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Step 5.3.3.1.2.1

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

$f (8 x^{3} - 216 x^{2} + 1944 x - 5832) = \frac{\sqrt[3]{2^{3} (x^{3} - 27 x^{2} + 243 x - 729)}}{2} + 9$

Step 5.3.3.1.2.2

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

$f (8 x^{3} - 216 x^{2} + 1944 x - 5832) = \frac{\sqrt[3]{2^{3} (x^{3} + {(- 3)}^{3} x^{2} + 243 x - 729)}}{2} + 9$

Step 5.3.3.1.3

Pull terms out from under the radical.

$f (8 x^{3} - 216 x^{2} + 1944 x - 5832) = \frac{2 \sqrt[3]{x^{3} + {(- 3)}^{3} x^{2} + 243 x - 729}}{2} + 9$

Step 5.3.3.1.4

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

$f (8 x^{3} - 216 x^{2} + 1944 x - 5832) = \frac{2 \sqrt[3]{x^{3} - 27 x^{2} + 243 x - 729}}{2} + 9$

Step 5.3.3.1.5

Rewrite $x^{3} - 27 x^{2} + 243 x - 729$ in a factored form.

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Step 5.3.3.1.5.1

Factor $x^{3} - 27 x^{2} + 243 x - 729$ using the rational roots test.

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Step 5.3.3.1.5.1.1

If a polynomial function has integer coefficients, then every rational zero will have the form $\frac{p}{q}$ where $p$ is a factor of the constant and $q$ is a factor of the leading coefficient.

$p = \pm 1, \pm 729, \pm 3, \pm 243, \pm 9, \pm 81, \pm 27$

$q = \pm 1$

Step 5.3.3.1.5.1.2

Find every combination of $\pm \frac{p}{q}$ . These are the possible roots of the polynomial function.

$\pm 1, \pm 729, \pm 3, \pm 243, \pm 9, \pm 81, \pm 27$

Step 5.3.3.1.5.1.3

Substitute $9$ and simplify the expression. In this case, the expression is equal to $0$ so $9$ is a root of the polynomial.

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Step 5.3.3.1.5.1.3.1

Substitute $9$ into the polynomial.

$9^{3} - 27 \cdot 9^{2} + 243 \cdot 9 - 729$

Step 5.3.3.1.5.1.3.2

Raise $9$ to the power of $3$ .

$729 - 27 \cdot 9^{2} + 243 \cdot 9 - 729$

Step 5.3.3.1.5.1.3.3

Raise $9$ to the power of $2$ .

$729 - 27 \cdot 81 + 243 \cdot 9 - 729$

Step 5.3.3.1.5.1.3.4

Multiply $- 27$ by $81$ .

$729 - 2187 + 243 \cdot 9 - 729$

Step 5.3.3.1.5.1.3.5

Subtract $2187$ from $729$ .

$- 1458 + 243 \cdot 9 - 729$

Step 5.3.3.1.5.1.3.6

Multiply $243$ by $9$ .

$- 1458 + 2187 - 729$

Step 5.3.3.1.5.1.3.7

Add $- 1458$ and $2187$ .

$729 - 729$

Step 5.3.3.1.5.1.3.8

Subtract $729$ from $729$ .

$0$

Step 5.3.3.1.5.1.4

Since $9$ is a known root, divide the polynomial by $x - 9$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{x^{3} - 27 x^{2} + 243 x - 729}{x - 9}$

Step 5.3.3.1.5.1.5

Divide $x^{3} - 27 x^{2} + 243 x - 729$ by $x - 9$ .

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Step 5.3.3.1.5.1.5.1

Set up the polynomials to be divided. If there is not a term for every exponent, insert one with a value of $0$ .

$x$

$9$

$x^{3}$

$27 x^{2}$

$243 x$

$729$

Step 5.3.3.1.5.1.5.2

Divide the highest order term in the dividend $x^{3}$ by the highest order term in divisor $x$ .

					$x^{2}$
	$x$	-	$9$		$x^{3}$	-	$27 x^{2}$	+	$243 x$	-	$729$

Step 5.3.3.1.5.1.5.3

Multiply the new quotient term by the divisor.

				$x^{2}$
$x$	-	$9$		$x^{3}$	-	$27 x^{2}$	+	$243 x$	-	$729$
			+	$x^{3}$	-	$9 x^{2}$

Step 5.3.3.1.5.1.5.4

The expression needs to be subtracted from the dividend, so change all the signs in $x^{3} - 9 x^{2}$

				$x^{2}$
$x$	-	$9$		$x^{3}$	-	$27 x^{2}$	+	$243 x$	-	$729$
			-	$x^{3}$	+	$9 x^{2}$

Step 5.3.3.1.5.1.5.5

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

				$x^{2}$
$x$	-	$9$		$x^{3}$	-	$27 x^{2}$	+	$243 x$	-	$729$
			-	$x^{3}$	+	$9 x^{2}$
					-	$18 x^{2}$

Step 5.3.3.1.5.1.5.6

Pull the next terms from the original dividend down into the current dividend.

				$x^{2}$
$x$	-	$9$		$x^{3}$	-	$27 x^{2}$	+	$243 x$	-	$729$
			-	$x^{3}$	+	$9 x^{2}$
					-	$18 x^{2}$	+	$243 x$

Step 5.3.3.1.5.1.5.7

Divide the highest order term in the dividend $- 18 x^{2}$ by the highest order term in divisor $x$ .

				$x^{2}$	-	$18 x$
$x$	-	$9$		$x^{3}$	-	$27 x^{2}$	+	$243 x$	-	$729$
			-	$x^{3}$	+	$9 x^{2}$
					-	$18 x^{2}$	+	$243 x$

Step 5.3.3.1.5.1.5.8

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$18 x$
$x$	-	$9$		$x^{3}$	-	$27 x^{2}$	+	$243 x$	-	$729$
			-	$x^{3}$	+	$9 x^{2}$
					-	$18 x^{2}$	+	$243 x$
					-	$18 x^{2}$	+	$162 x$

Step 5.3.3.1.5.1.5.9

The expression needs to be subtracted from the dividend, so change all the signs in $- 18 x^{2} + 162 x$

				$x^{2}$	-	$18 x$
$x$	-	$9$		$x^{3}$	-	$27 x^{2}$	+	$243 x$	-	$729$
			-	$x^{3}$	+	$9 x^{2}$
					-	$18 x^{2}$	+	$243 x$
					+	$18 x^{2}$	-	$162 x$

Step 5.3.3.1.5.1.5.10

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

				$x^{2}$	-	$18 x$
$x$	-	$9$		$x^{3}$	-	$27 x^{2}$	+	$243 x$	-	$729$
			-	$x^{3}$	+	$9 x^{2}$
					-	$18 x^{2}$	+	$243 x$
					+	$18 x^{2}$	-	$162 x$
							+	$81 x$

Step 5.3.3.1.5.1.5.11

Pull the next terms from the original dividend down into the current dividend.

				$x^{2}$	-	$18 x$
$x$	-	$9$		$x^{3}$	-	$27 x^{2}$	+	$243 x$	-	$729$
			-	$x^{3}$	+	$9 x^{2}$
					-	$18 x^{2}$	+	$243 x$
					+	$18 x^{2}$	-	$162 x$
							+	$81 x$	-	$729$

Step 5.3.3.1.5.1.5.12

Divide the highest order term in the dividend $81 x$ by the highest order term in divisor $x$ .

				$x^{2}$	-	$18 x$	+	$81$
$x$	-	$9$		$x^{3}$	-	$27 x^{2}$	+	$243 x$	-	$729$
			-	$x^{3}$	+	$9 x^{2}$
					-	$18 x^{2}$	+	$243 x$
					+	$18 x^{2}$	-	$162 x$
							+	$81 x$	-	$729$

Step 5.3.3.1.5.1.5.13

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$18 x$	+	$81$
$x$	-	$9$		$x^{3}$	-	$27 x^{2}$	+	$243 x$	-	$729$
			-	$x^{3}$	+	$9 x^{2}$
					-	$18 x^{2}$	+	$243 x$
					+	$18 x^{2}$	-	$162 x$
							+	$81 x$	-	$729$
							+	$81 x$	-	$729$

Step 5.3.3.1.5.1.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $81 x - 729$

				$x^{2}$	-	$18 x$	+	$81$
$x$	-	$9$		$x^{3}$	-	$27 x^{2}$	+	$243 x$	-	$729$
			-	$x^{3}$	+	$9 x^{2}$
					-	$18 x^{2}$	+	$243 x$
					+	$18 x^{2}$	-	$162 x$
							+	$81 x$	-	$729$
							-	$81 x$	+	$729$

Step 5.3.3.1.5.1.5.15

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

				$x^{2}$	-	$18 x$	+	$81$
$x$	-	$9$		$x^{3}$	-	$27 x^{2}$	+	$243 x$	-	$729$
			-	$x^{3}$	+	$9 x^{2}$
					-	$18 x^{2}$	+	$243 x$
					+	$18 x^{2}$	-	$162 x$
							+	$81 x$	-	$729$
							-	$81 x$	+	$729$
										$0$

Step 5.3.3.1.5.1.5.16

Since the remander is $0$ , the final answer is the quotient.

$x^{2} - 18 x + 81$

Step 5.3.3.1.5.1.6

Write $x^{3} - 27 x^{2} + 243 x - 729$ as a set of factors.

$f (8 x^{3} - 216 x^{2} + 1944 x - 5832) = \frac{2 \sqrt[3]{(x - 9) (x^{2} - 18 x + 81)}}{2} + 9$

Step 5.3.3.1.5.2

Factor using the perfect square rule.

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Step 5.3.3.1.5.2.1

Rewrite $81$ as $9^{2}$ .

$f (8 x^{3} - 216 x^{2} + 1944 x - 5832) = \frac{2 \sqrt[3]{(x - 9) (x^{2} - 18 x + 9^{2})}}{2} + 9$

Step 5.3.3.1.5.2.2

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

$18 x = 2 \cdot x \cdot 9$

Step 5.3.3.1.5.2.3

Rewrite the polynomial.

$f (8 x^{3} - 216 x^{2} + 1944 x - 5832) = \frac{2 \sqrt[3]{(x - 9) (x^{2} - 2 \cdot x \cdot 9 + 9^{2})}}{2} + 9$

Step 5.3.3.1.5.2.4

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = x$ and $b = 9$ .

$f (8 x^{3} - 216 x^{2} + 1944 x - 5832) = \frac{2 \sqrt[3]{(x - 9) {(x - 9)}^{2}}}{2} + 9$

Step 5.3.3.1.5.3

Combine like factors.

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Step 5.3.3.1.5.3.1

Raise $x - 9$ to the power of $1$ .

$f (8 x^{3} - 216 x^{2} + 1944 x - 5832) = \frac{2 \sqrt[3]{(x - 9) {(x - 9)}^{2}}}{2} + 9$

Step 5.3.3.1.5.3.2

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

$f (8 x^{3} - 216 x^{2} + 1944 x - 5832) = \frac{2 \sqrt[3]{{(x - 9)}^{1 + 2}}}{2} + 9$

Step 5.3.3.1.5.3.3

Add $1$ and $2$ .

$f (8 x^{3} - 216 x^{2} + 1944 x - 5832) = \frac{2 \sqrt[3]{{(x - 9)}^{3}}}{2} + 9$

Step 5.3.3.1.6

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

$f (8 x^{3} - 216 x^{2} + 1944 x - 5832) = \frac{2 (x - 9)}{2} + 9$

Step 5.3.3.1.7

Apply the distributive property.

$f (8 x^{3} - 216 x^{2} + 1944 x - 5832) = \frac{2 x + 2 \cdot - 9}{2} + 9$

Step 5.3.3.1.8

Multiply $2$ by $- 9$ .

$f (8 x^{3} - 216 x^{2} + 1944 x - 5832) = \frac{2 x - 18}{2} + 9$

Step 5.3.3.1.9

Factor $2$ out of $2 x - 18$ .

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Step 5.3.3.1.9.1

Factor $2$ out of $2 x$ .

$f (8 x^{3} - 216 x^{2} + 1944 x - 5832) = \frac{2 (x) - 18}{2} + 9$

Step 5.3.3.1.9.2

Factor $2$ out of $- 18$ .

$f (8 x^{3} - 216 x^{2} + 1944 x - 5832) = \frac{2 x + 2 \cdot - 9}{2} + 9$

Step 5.3.3.1.9.3

Factor $2$ out of $2 x + 2 \cdot - 9$ .

$f (8 x^{3} - 216 x^{2} + 1944 x - 5832) = \frac{2 (x - 9)}{2} + 9$

Step 5.3.3.2

Cancel the common factor of $2$ .

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Step 5.3.3.2.1

Cancel the common factor.

$f (8 x^{3} - 216 x^{2} + 1944 x - 5832) = \frac{2 (x - 9)}{2} + 9$

Step 5.3.3.2.2

Divide $x - 9$ by $1$ .

$f (8 x^{3} - 216 x^{2} + 1944 x - 5832) = x - 9 + 9$

Step 5.3.4

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

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Step 5.3.4.1

Add $- 9$ and $9$ .

$f (8 x^{3} - 216 x^{2} + 1944 x - 5832) = x + 0$

Step 5.3.4.2

Add $x$ and $0$ .

$f (8 x^{3} - 216 x^{2} + 1944 x - 5832) = x$

Step 5.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = 8 x^{3} - 216 x^{2} + 1944 x - 5832$ is the inverse of $f (x) = \frac{\sqrt[3]{x}}{2} + 9$ .

$f^{- 1} (x) = 8 x^{3} - 216 x^{2} + 1944 x - 5832$