Precalculus Examples

$f (x) = \frac{x^{\frac{1}{3}}}{6} + 7$ ; find $f^{- 1} (x)$

Step 1

Write $f (x) = \frac{x^{\frac{1}{3}}}{6} + 7$ as an equation.

$y = \frac{x^{\frac{1}{3}}}{6} + 7$

Step 2

Interchange the variables.

$x = \frac{y^{\frac{1}{3}}}{6} + 7$

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $\frac{y^{\frac{1}{3}}}{6} + 7 = x$ .

$\frac{y^{\frac{1}{3}}}{6} + 7 = x$

Step 3.2

Move all terms not containing $y$ to the right side of the equation.

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Step 3.2.1

Subtract $7$ from both sides of the equation.

$\frac{y^{\frac{1}{3}}}{6} - x = - 7$

Step 3.2.2

Add $x$ to both sides of the equation.

$\frac{y^{\frac{1}{3}}}{6} = - 7 + x$

Step 3.3

Multiply both sides of the equation by $6$ .

$6 \frac{y^{\frac{1}{3}}}{6} = 6 (- 7 + x)$

Step 3.4

Simplify both sides of the equation.

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Step 3.4.1

Simplify the left side.

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Step 3.4.1.1

Cancel the common factor of $6$ .

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Step 3.4.1.1.1

Cancel the common factor.

$6 \frac{y^{\frac{1}{3}}}{6} = 6 (- 7 + x)$

Step 3.4.1.1.2

Rewrite the expression.

$y^{\frac{1}{3}} = 6 (- 7 + x)$

Step 3.4.2

Simplify the right side.

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Step 3.4.2.1

Simplify $6 (- 7 + x)$ .

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Step 3.4.2.1.1

Apply the distributive property.

$y^{\frac{1}{3}} = 6 \cdot - 7 + 6 x$

Step 3.4.2.1.2

Multiply $6$ by $- 7$ .

$y^{\frac{1}{3}} = - 42 + 6 x$

Step 3.5

Raise each side of the equation to the power of $3$ to eliminate the fractional exponent on the left side.

${(y^{\frac{1}{3}})}^{3} = {(- 42 + 6 x)}^{3}$

Step 3.6

Simplify the exponent.

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Step 3.6.1

Simplify the left side.

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Step 3.6.1.1

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

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Step 3.6.1.1.1

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

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Step 3.6.1.1.1.1

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

$y^{\frac{1}{3} \cdot 3} = {(- 42 + 6 x)}^{3}$

Step 3.6.1.1.1.2

Cancel the common factor of $3$ .

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Step 3.6.1.1.1.2.1

Cancel the common factor.

$y^{\frac{1}{3} \cdot 3} = {(- 42 + 6 x)}^{3}$

Step 3.6.1.1.1.2.2

Rewrite the expression.

$y^{1} = {(- 42 + 6 x)}^{3}$

Step 3.6.1.1.2

Simplify.

$y = {(- 42 + 6 x)}^{3}$

Step 3.6.2

Simplify the right side.

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Step 3.6.2.1

Simplify ${(- 42 + 6 x)}^{3}$ .

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Step 3.6.2.1.1

Use the Binomial Theorem.

$y = {(- 42)}^{3} + 3 {(- 42)}^{2} (6 x) + 3 \cdot - 42 {(6 x)}^{2} + {(6 x)}^{3}$

Step 3.6.2.1.2

Simplify each term.

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Step 3.6.2.1.2.1

Raise $- 42$ to the power of $3$ .

$y = - 74088 + 3 {(- 42)}^{2} (6 x) + 3 \cdot - 42 {(6 x)}^{2} + {(6 x)}^{3}$

Step 3.6.2.1.2.2

Raise $- 42$ to the power of $2$ .

$y = - 74088 + 3 \cdot 1764 (6 x) + 3 \cdot - 42 {(6 x)}^{2} + {(6 x)}^{3}$

Step 3.6.2.1.2.3

Multiply $3$ by $1764$ .

$y = - 74088 + 5292 (6 x) + 3 \cdot - 42 {(6 x)}^{2} + {(6 x)}^{3}$

Step 3.6.2.1.2.4

Multiply $6$ by $5292$ .

$y = - 74088 + 31752 x + 3 \cdot - 42 {(6 x)}^{2} + {(6 x)}^{3}$

Step 3.6.2.1.2.5

Multiply $3$ by $- 42$ .

$y = - 74088 + 31752 x - 126 {(6 x)}^{2} + {(6 x)}^{3}$

Step 3.6.2.1.2.6

Apply the product rule to $6 x$ .

$y = - 74088 + 31752 x - 126 (6^{2} x^{2}) + {(6 x)}^{3}$

Step 3.6.2.1.2.7

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

$y = - 74088 + 31752 x - 126 (36 x^{2}) + {(6 x)}^{3}$

Step 3.6.2.1.2.8

Multiply $36$ by $- 126$ .

$y = - 74088 + 31752 x - 4536 x^{2} + {(6 x)}^{3}$

Step 3.6.2.1.2.9

Apply the product rule to $6 x$ .

$y = - 74088 + 31752 x - 4536 x^{2} + 6^{3} x^{3}$

Step 3.6.2.1.2.10

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

$y = - 74088 + 31752 x - 4536 x^{2} + 216 x^{3}$

Step 3.7

Simplify $- 74088 + 31752 x - 4536 x^{2} + 216 x^{3}$ .

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Step 3.7.1

Move $- 74088$ .

$y = 31752 x - 4536 x^{2} + 216 x^{3} - 74088$

Step 3.7.2

Move $31752 x$ .

$y = - 4536 x^{2} + 216 x^{3} + 31752 x - 74088$

Step 3.7.3

Reorder $- 4536 x^{2}$ and $216 x^{3}$ .

$y = 216 x^{3} - 4536 x^{2} + 31752 x - 74088$

Step 4

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

$f^{- 1} (x) = 216 x^{3} - 4536 x^{2} + 31752 x - 74088$

Step 5

Verify if $f^{- 1} (x) = 216 x^{3} - 4536 x^{2} + 31752 x - 74088$ is the inverse of $f (x) = \frac{x^{\frac{1}{3}}}{6} + 7$ .

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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 $f^{- 1} (\frac{x^{\frac{1}{3}}}{6} + 7)$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\frac{x^{\frac{1}{3}}}{6} + 7) = 216 {(\frac{x^{\frac{1}{3}}}{6} + 7)}^{3} - 4536 {(\frac{x^{\frac{1}{3}}}{6} + 7)}^{2} + 31752 (\frac{x^{\frac{1}{3}}}{6} + 7) - 74088$

Step 5.2.3

Simplify each term.

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Step 5.2.3.1

Use the Binomial Theorem.

$f^{- 1} (\frac{x^{\frac{1}{3}}}{6} + 7) = 216 ({(\frac{x^{\frac{1}{3}}}{6})}^{3} + 3 {(\frac{x^{\frac{1}{3}}}{6})}^{2} \cdot 7 + 3 (\frac{x^{\frac{1}{3}}}{6}) \cdot 7^{2} + 7^{3}) - 4536 {(\frac{x^{\frac{1}{3}}}{6} + 7)}^{2} + 31752 (\frac{x^{\frac{1}{3}}}{6} + 7) - 74088$

Step 5.2.3.2

Simplify each term.

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Step 5.2.3.2.1

Apply the product rule to $\frac{x^{\frac{1}{3}}}{6}$ .

$f^{- 1} (\frac{x^{\frac{1}{3}}}{6} + 7) = 216 (\frac{{(x^{\frac{1}{3}})}^{3}}{6^{3}} + 3 {(\frac{x^{\frac{1}{3}}}{6})}^{2} \cdot 7 + 3 (\frac{x^{\frac{1}{3}}}{6}) \cdot 7^{2} + 7^{3}) - 4536 {(\frac{x^{\frac{1}{3}}}{6} + 7)}^{2} + 31752 (\frac{x^{\frac{1}{3}}}{6} + 7) - 74088$

Step 5.2.3.2.2

Simplify the numerator.

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Step 5.2.3.2.2.1

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

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Step 5.2.3.2.2.1.1

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

$f^{- 1} (\frac{x^{\frac{1}{3}}}{6} + 7) = 216 (\frac{x^{\frac{1}{3} \cdot 3}}{6^{3}} + 3 {(\frac{x^{\frac{1}{3}}}{6})}^{2} \cdot 7 + 3 (\frac{x^{\frac{1}{3}}}{6}) \cdot 7^{2} + 7^{3}) - 4536 {(\frac{x^{\frac{1}{3}}}{6} + 7)}^{2} + 31752 (\frac{x^{\frac{1}{3}}}{6} + 7) - 74088$

Step 5.2.3.2.2.1.2

Cancel the common factor of $3$ .

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Step 5.2.3.2.2.1.2.1

Cancel the common factor.

Step 5.2.3.2.2.1.2.2

Rewrite the expression.

$f^{- 1} (\frac{x^{\frac{1}{3}}}{6} + 7) = 216 (\frac{x}{6^{3}} + 3 {(\frac{x^{\frac{1}{3}}}{6})}^{2} \cdot 7 + 3 (\frac{x^{\frac{1}{3}}}{6}) \cdot 7^{2} + 7^{3}) - 4536 {(\frac{x^{\frac{1}{3}}}{6} + 7)}^{2} + 31752 (\frac{x^{\frac{1}{3}}}{6} + 7) - 74088$

Step 5.2.3.2.2.2

Simplify.

Step 5.2.3.2.3

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

$f^{- 1} (\frac{x^{\frac{1}{3}}}{6} + 7) = 216 (\frac{x}{216} + 3 {(\frac{x^{\frac{1}{3}}}{6})}^{2} \cdot 7 + 3 (\frac{x^{\frac{1}{3}}}{6}) \cdot 7^{2} + 7^{3}) - 4536 {(\frac{x^{\frac{1}{3}}}{6} + 7)}^{2} + 31752 (\frac{x^{\frac{1}{3}}}{6} + 7) - 74088$

Step 5.2.3.2.4

Apply the product rule to $\frac{x^{\frac{1}{3}}}{6}$ .

$f^{- 1} (\frac{x^{\frac{1}{3}}}{6} + 7) = 216 (\frac{x}{216} + 3 (\frac{{(x^{\frac{1}{3}})}^{2}}{6^{2}}) \cdot 7 + 3 (\frac{x^{\frac{1}{3}}}{6}) \cdot 7^{2} + 7^{3}) - 4536 {(\frac{x^{\frac{1}{3}}}{6} + 7)}^{2} + 31752 (\frac{x^{\frac{1}{3}}}{6} + 7) - 74088$

Step 5.2.3.2.5

Multiply the exponents in ${(x^{\frac{1}{3}})}^{2}$ .

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Step 5.2.3.2.5.1

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

$f^{- 1} (\frac{x^{\frac{1}{3}}}{6} + 7) = 216 (\frac{x}{216} + 3 (\frac{x^{\frac{1}{3} \cdot 2}}{6^{2}}) \cdot 7 + 3 (\frac{x^{\frac{1}{3}}}{6}) \cdot 7^{2} + 7^{3}) - 4536 {(\frac{x^{\frac{1}{3}}}{6} + 7)}^{2} + 31752 (\frac{x^{\frac{1}{3}}}{6} + 7) - 74088$

Step 5.2.3.2.5.2

Combine $\frac{1}{3}$ and $2$ .

$f^{- 1} (\frac{x^{\frac{1}{3}}}{6} + 7) = 216 (\frac{x}{216} + 3 (\frac{x^{\frac{2}{3}}}{6^{2}}) \cdot 7 + 3 (\frac{x^{\frac{1}{3}}}{6}) \cdot 7^{2} + 7^{3}) - 4536 {(\frac{x^{\frac{1}{3}}}{6} + 7)}^{2} + 31752 (\frac{x^{\frac{1}{3}}}{6} + 7) - 74088$

Step 5.2.3.2.6

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

$f^{- 1} (\frac{x^{\frac{1}{3}}}{6} + 7) = 216 (\frac{x}{216} + 3 (\frac{x^{\frac{2}{3}}}{36}) \cdot 7 + 3 (\frac{x^{\frac{1}{3}}}{6}) \cdot 7^{2} + 7^{3}) - 4536 {(\frac{x^{\frac{1}{3}}}{6} + 7)}^{2} + 31752 (\frac{x^{\frac{1}{3}}}{6} + 7) - 74088$

Step 5.2.3.2.7

Cancel the common factor of $3$ .

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Step 5.2.3.2.7.1

Factor $3$ out of $36$ .

$f^{- 1} (\frac{x^{\frac{1}{3}}}{6} + 7) = 216 (\frac{x}{216} + 3 (\frac{x^{\frac{2}{3}}}{3 (12)}) \cdot 7 + 3 (\frac{x^{\frac{1}{3}}}{6}) \cdot 7^{2} + 7^{3}) - 4536 {(\frac{x^{\frac{1}{3}}}{6} + 7)}^{2} + 31752 (\frac{x^{\frac{1}{3}}}{6} + 7) - 74088$

Step 5.2.3.2.7.2

Cancel the common factor.

$f^{- 1} (\frac{x^{\frac{1}{3}}}{6} + 7) = 216 (\frac{x}{216} + 3 (\frac{x^{\frac{2}{3}}}{3 \cdot 12}) \cdot 7 + 3 (\frac{x^{\frac{1}{3}}}{6}) \cdot 7^{2} + 7^{3}) - 4536 {(\frac{x^{\frac{1}{3}}}{6} + 7)}^{2} + 31752 (\frac{x^{\frac{1}{3}}}{6} + 7) - 74088$

Step 5.2.3.2.7.3

Rewrite the expression.

$f^{- 1} (\frac{x^{\frac{1}{3}}}{6} + 7) = 216 (\frac{x}{216} + \frac{x^{\frac{2}{3}}}{12} \cdot 7 + 3 (\frac{x^{\frac{1}{3}}}{6}) \cdot 7^{2} + 7^{3}) - 4536 {(\frac{x^{\frac{1}{3}}}{6} + 7)}^{2} + 31752 (\frac{x^{\frac{1}{3}}}{6} + 7) - 74088$

Step 5.2.3.2.8

Combine $\frac{x^{\frac{2}{3}}}{12}$ and $7$ .

$f^{- 1} (\frac{x^{\frac{1}{3}}}{6} + 7) = 216 (\frac{x}{216} + \frac{x^{\frac{2}{3}} \cdot 7}{12} + 3 (\frac{x^{\frac{1}{3}}}{6}) \cdot 7^{2} + 7^{3}) - 4536 {(\frac{x^{\frac{1}{3}}}{6} + 7)}^{2} + 31752 (\frac{x^{\frac{1}{3}}}{6} + 7) - 74088$

Step 5.2.3.2.9

Move $7$ to the left of $x^{\frac{2}{3}}$ .

$f^{- 1} (\frac{x^{\frac{1}{3}}}{6} + 7) = 216 (\frac{x}{216} + \frac{7 \cdot x^{\frac{2}{3}}}{12} + 3 (\frac{x^{\frac{1}{3}}}{6}) \cdot 7^{2} + 7^{3}) - 4536 {(\frac{x^{\frac{1}{3}}}{6} + 7)}^{2} + 31752 (\frac{x^{\frac{1}{3}}}{6} + 7) - 74088$

Step 5.2.3.2.10

Cancel the common factor of $3$ .

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Step 5.2.3.2.10.1

Factor $3$ out of $6$ .

$f^{- 1} (\frac{x^{\frac{1}{3}}}{6} + 7) = 216 (\frac{x}{216} + \frac{7 x^{\frac{2}{3}}}{12} + 3 (\frac{x^{\frac{1}{3}}}{3 (2)}) \cdot 7^{2} + 7^{3}) - 4536 {(\frac{x^{\frac{1}{3}}}{6} + 7)}^{2} + 31752 (\frac{x^{\frac{1}{3}}}{6} + 7) - 74088$

Step 5.2.3.2.10.2

Cancel the common factor.

$f^{- 1} (\frac{x^{\frac{1}{3}}}{6} + 7) = 216 (\frac{x}{216} + \frac{7 x^{\frac{2}{3}}}{12} + 3 (\frac{x^{\frac{1}{3}}}{3 \cdot 2}) \cdot 7^{2} + 7^{3}) - 4536 {(\frac{x^{\frac{1}{3}}}{6} + 7)}^{2} + 31752 (\frac{x^{\frac{1}{3}}}{6} + 7) - 74088$

Step 5.2.3.2.10.3

Rewrite the expression.

$f^{- 1} (\frac{x^{\frac{1}{3}}}{6} + 7) = 216 (\frac{x}{216} + \frac{7 x^{\frac{2}{3}}}{12} + \frac{x^{\frac{1}{3}}}{2} \cdot 7^{2} + 7^{3}) - 4536 {(\frac{x^{\frac{1}{3}}}{6} + 7)}^{2} + 31752 (\frac{x^{\frac{1}{3}}}{6} + 7) - 74088$

Step 5.2.3.2.11

Raise $7$ to the power of $2$ .

$f^{- 1} (\frac{x^{\frac{1}{3}}}{6} + 7) = 216 (\frac{x}{216} + \frac{7 x^{\frac{2}{3}}}{12} + \frac{x^{\frac{1}{3}}}{2} \cdot 49 + 7^{3}) - 4536 {(\frac{x^{\frac{1}{3}}}{6} + 7)}^{2} + 31752 (\frac{x^{\frac{1}{3}}}{6} + 7) - 74088$

Step 5.2.3.2.12

Combine $\frac{x^{\frac{1}{3}}}{2}$ and $49$ .

$f^{- 1} (\frac{x^{\frac{1}{3}}}{6} + 7) = 216 (\frac{x}{216} + \frac{7 x^{\frac{2}{3}}}{12} + \frac{x^{\frac{1}{3}} \cdot 49}{2} + 7^{3}) - 4536 {(\frac{x^{\frac{1}{3}}}{6} + 7)}^{2} + 31752 (\frac{x^{\frac{1}{3}}}{6} + 7) - 74088$

Step 5.2.3.2.13

Move $49$ to the left of $x^{\frac{1}{3}}$ .

$f^{- 1} (\frac{x^{\frac{1}{3}}}{6} + 7) = 216 (\frac{x}{216} + \frac{7 x^{\frac{2}{3}}}{12} + \frac{49 \cdot x^{\frac{1}{3}}}{2} + 7^{3}) - 4536 {(\frac{x^{\frac{1}{3}}}{6} + 7)}^{2} + 31752 (\frac{x^{\frac{1}{3}}}{6} + 7) - 74088$

Step 5.2.3.2.14

Raise $7$ to the power of $3$ .

$f^{- 1} (\frac{x^{\frac{1}{3}}}{6} + 7) = 216 (\frac{x}{216} + \frac{7 x^{\frac{2}{3}}}{12} + \frac{49 x^{\frac{1}{3}}}{2} + 343) - 4536 {(\frac{x^{\frac{1}{3}}}{6} + 7)}^{2} + 31752 (\frac{x^{\frac{1}{3}}}{6} + 7) - 74088$

Step 5.2.3.3

Apply the distributive property.

$f^{- 1} (\frac{x^{\frac{1}{3}}}{6} + 7) = 216 (\frac{x}{216}) + 216 (\frac{7 x^{\frac{2}{3}}}{12}) + 216 (\frac{49 x^{\frac{1}{3}}}{2}) + 216 \cdot 343 - 4536 {(\frac{x^{\frac{1}{3}}}{6} + 7)}^{2} + 31752 (\frac{x^{\frac{1}{3}}}{6} + 7) - 74088$

Step 5.2.3.4

Simplify.

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Step 5.2.3.4.1

Cancel the common factor of $216$ .

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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.

$f^{- 1} (\frac{x^{\frac{1}{3}}}{6} + 7) = x + 216 (\frac{7 x^{\frac{2}{3}}}{12}) + 216 (\frac{49 x^{\frac{1}{3}}}{2}) + 216 \cdot 343 - 4536 {(\frac{x^{\frac{1}{3}}}{6} + 7)}^{2} + 31752 (\frac{x^{\frac{1}{3}}}{6} + 7) - 74088$

Step 5.2.3.4.2

Cancel the common factor of $12$ .

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Step 5.2.3.4.2.1

Factor $12$ out of $216$ .

$f^{- 1} (\frac{x^{\frac{1}{3}}}{6} + 7) = x + 12 (18) (\frac{7 x^{\frac{2}{3}}}{12}) + 216 (\frac{49 x^{\frac{1}{3}}}{2}) + 216 \cdot 343 - 4536 {(\frac{x^{\frac{1}{3}}}{6} + 7)}^{2} + 31752 (\frac{x^{\frac{1}{3}}}{6} + 7) - 74088$

Step 5.2.3.4.2.2

Cancel the common factor.

$f^{- 1} (\frac{x^{\frac{1}{3}}}{6} + 7) = x + 12 \cdot (18 (\frac{7 x^{\frac{2}{3}}}{12})) + 216 (\frac{49 x^{\frac{1}{3}}}{2}) + 216 \cdot 343 - 4536 {(\frac{x^{\frac{1}{3}}}{6} + 7)}^{2} + 31752 (\frac{x^{\frac{1}{3}}}{6} + 7) - 74088$

Step 5.2.3.4.2.3

Rewrite the expression.

$f^{- 1} (\frac{x^{\frac{1}{3}}}{6} + 7) = x + 18 (7 x^{\frac{2}{3}}) + 216 (\frac{49 x^{\frac{1}{3}}}{2}) + 216 \cdot 343 - 4536 {(\frac{x^{\frac{1}{3}}}{6} + 7)}^{2} + 31752 (\frac{x^{\frac{1}{3}}}{6} + 7) - 74088$

Step 5.2.3.4.3

Multiply $7$ by $18$ .

$f^{- 1} (\frac{x^{\frac{1}{3}}}{6} + 7) = x + 126 x^{\frac{2}{3}} + 216 (\frac{49 x^{\frac{1}{3}}}{2}) + 216 \cdot 343 - 4536 {(\frac{x^{\frac{1}{3}}}{6} + 7)}^{2} + 31752 (\frac{x^{\frac{1}{3}}}{6} + 7) - 74088$

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 $216$ .

$f^{- 1} (\frac{x^{\frac{1}{3}}}{6} + 7) = x + 126 x^{\frac{2}{3}} + 2 (108) (\frac{49 x^{\frac{1}{3}}}{2}) + 216 \cdot 343 - 4536 {(\frac{x^{\frac{1}{3}}}{6} + 7)}^{2} + 31752 (\frac{x^{\frac{1}{3}}}{6} + 7) - 74088$

Step 5.2.3.4.4.2

Cancel the common factor.

$f^{- 1} (\frac{x^{\frac{1}{3}}}{6} + 7) = x + 126 x^{\frac{2}{3}} + 2 \cdot (108 (\frac{49 x^{\frac{1}{3}}}{2})) + 216 \cdot 343 - 4536 {(\frac{x^{\frac{1}{3}}}{6} + 7)}^{2} + 31752 (\frac{x^{\frac{1}{3}}}{6} + 7) - 74088$

Step 5.2.3.4.4.3

Rewrite the expression.

$f^{- 1} (\frac{x^{\frac{1}{3}}}{6} + 7) = x + 126 x^{\frac{2}{3}} + 108 (49 x^{\frac{1}{3}}) + 216 \cdot 343 - 4536 {(\frac{x^{\frac{1}{3}}}{6} + 7)}^{2} + 31752 (\frac{x^{\frac{1}{3}}}{6} + 7) - 74088$

Step 5.2.3.4.5

Multiply $49$ by $108$ .

$f^{- 1} (\frac{x^{\frac{1}{3}}}{6} + 7) = x + 126 x^{\frac{2}{3}} + 5292 x^{\frac{1}{3}} + 216 \cdot 343 - 4536 {(\frac{x^{\frac{1}{3}}}{6} + 7)}^{2} + 31752 (\frac{x^{\frac{1}{3}}}{6} + 7) - 74088$

Step 5.2.3.4.6

Multiply $216$ by $343$ .

$f^{- 1} (\frac{x^{\frac{1}{3}}}{6} + 7) = x + 126 x^{\frac{2}{3}} + 5292 x^{\frac{1}{3}} + 74088 - 4536 {(\frac{x^{\frac{1}{3}}}{6} + 7)}^{2} + 31752 (\frac{x^{\frac{1}{3}}}{6} + 7) - 74088$

Step 5.2.3.5

Rewrite ${(\frac{x^{\frac{1}{3}}}{6} + 7)}^{2}$ as $(\frac{x^{\frac{1}{3}}}{6} + 7) (\frac{x^{\frac{1}{3}}}{6} + 7)$ .

$f^{- 1} (\frac{x^{\frac{1}{3}}}{6} + 7) = x + 126 x^{\frac{2}{3}} + 5292 x^{\frac{1}{3}} + 74088 - 4536 ((\frac{x^{\frac{1}{3}}}{6} + 7) (\frac{x^{\frac{1}{3}}}{6} + 7)) + 31752 (\frac{x^{\frac{1}{3}}}{6} + 7) - 74088$

Step 5.2.3.6

Expand $(\frac{x^{\frac{1}{3}}}{6} + 7) (\frac{x^{\frac{1}{3}}}{6} + 7)$ using the FOIL Method.

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Step 5.2.3.6.1

Apply the distributive property.

$f^{- 1} (\frac{x^{\frac{1}{3}}}{6} + 7) = x + 126 x^{\frac{2}{3}} + 5292 x^{\frac{1}{3}} + 74088 - 4536 (\frac{x^{\frac{1}{3}}}{6} \cdot (\frac{x^{\frac{1}{3}}}{6} + 7) + 7 (\frac{x^{\frac{1}{3}}}{6} + 7)) + 31752 (\frac{x^{\frac{1}{3}}}{6} + 7) - 74088$

Step 5.2.3.6.2

Apply the distributive property.

$f^{- 1} (\frac{x^{\frac{1}{3}}}{6} + 7) = x + 126 x^{\frac{2}{3}} + 5292 x^{\frac{1}{3}} + 74088 - 4536 (\frac{x^{\frac{1}{3}}}{6} \cdot \frac{x^{\frac{1}{3}}}{6} + \frac{x^{\frac{1}{3}}}{6} \cdot 7 + 7 (\frac{x^{\frac{1}{3}}}{6} + 7)) + 31752 (\frac{x^{\frac{1}{3}}}{6} + 7) - 74088$

Step 5.2.3.6.3

Apply the distributive property.

$f^{- 1} (\frac{x^{\frac{1}{3}}}{6} + 7) = x + 126 x^{\frac{2}{3}} + 5292 x^{\frac{1}{3}} + 74088 - 4536 (\frac{x^{\frac{1}{3}}}{6} \cdot \frac{x^{\frac{1}{3}}}{6} + \frac{x^{\frac{1}{3}}}{6} \cdot 7 + 7 (\frac{x^{\frac{1}{3}}}{6}) + 7 \cdot 7) + 31752 (\frac{x^{\frac{1}{3}}}{6} + 7) - 74088$

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

Combine.

$f^{- 1} (\frac{x^{\frac{1}{3}}}{6} + 7) = x + 126 x^{\frac{2}{3}} + 5292 x^{\frac{1}{3}} + 74088 - 4536 (\frac{x^{\frac{1}{3}} x^{\frac{1}{3}}}{6 \cdot 6} + \frac{x^{\frac{1}{3}}}{6} \cdot 7 + 7 (\frac{x^{\frac{1}{3}}}{6}) + 7 \cdot 7) + 31752 (\frac{x^{\frac{1}{3}}}{6} + 7) - 74088$

Step 5.2.3.7.1.2

Multiply $x^{\frac{1}{3}}$ by $x^{\frac{1}{3}}$ by adding the exponents.

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Step 5.2.3.7.1.2.1

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

$f^{- 1} (\frac{x^{\frac{1}{3}}}{6} + 7) = x + 126 x^{\frac{2}{3}} + 5292 x^{\frac{1}{3}} + 74088 - 4536 (\frac{x^{\frac{1}{3} + \frac{1}{3}}}{6 \cdot 6} + \frac{x^{\frac{1}{3}}}{6} \cdot 7 + 7 (\frac{x^{\frac{1}{3}}}{6}) + 7 \cdot 7) + 31752 (\frac{x^{\frac{1}{3}}}{6} + 7) - 74088$

Step 5.2.3.7.1.2.2

Combine the numerators over the common denominator.

$f^{- 1} (\frac{x^{\frac{1}{3}}}{6} + 7) = x + 126 x^{\frac{2}{3}} + 5292 x^{\frac{1}{3}} + 74088 - 4536 (\frac{x^{\frac{1 + 1}{3}}}{6 \cdot 6} + \frac{x^{\frac{1}{3}}}{6} \cdot 7 + 7 (\frac{x^{\frac{1}{3}}}{6}) + 7 \cdot 7) + 31752 (\frac{x^{\frac{1}{3}}}{6} + 7) - 74088$

Step 5.2.3.7.1.2.3

Add $1$ and $1$ .

$f^{- 1} (\frac{x^{\frac{1}{3}}}{6} + 7) = x + 126 x^{\frac{2}{3}} + 5292 x^{\frac{1}{3}} + 74088 - 4536 (\frac{x^{\frac{2}{3}}}{6 \cdot 6} + \frac{x^{\frac{1}{3}}}{6} \cdot 7 + 7 (\frac{x^{\frac{1}{3}}}{6}) + 7 \cdot 7) + 31752 (\frac{x^{\frac{1}{3}}}{6} + 7) - 74088$

Step 5.2.3.7.1.3

Multiply $6$ by $6$ .

$f^{- 1} (\frac{x^{\frac{1}{3}}}{6} + 7) = x + 126 x^{\frac{2}{3}} + 5292 x^{\frac{1}{3}} + 74088 - 4536 (\frac{x^{\frac{2}{3}}}{36} + \frac{x^{\frac{1}{3}}}{6} \cdot 7 + 7 (\frac{x^{\frac{1}{3}}}{6}) + 7 \cdot 7) + 31752 (\frac{x^{\frac{1}{3}}}{6} + 7) - 74088$

Step 5.2.3.7.1.4

Combine $\frac{x^{\frac{1}{3}}}{6}$ and $7$ .

$f^{- 1} (\frac{x^{\frac{1}{3}}}{6} + 7) = x + 126 x^{\frac{2}{3}} + 5292 x^{\frac{1}{3}} + 74088 - 4536 (\frac{x^{\frac{2}{3}}}{36} + \frac{x^{\frac{1}{3}} \cdot 7}{6} + 7 (\frac{x^{\frac{1}{3}}}{6}) + 7 \cdot 7) + 31752 (\frac{x^{\frac{1}{3}}}{6} + 7) - 74088$

Step 5.2.3.7.1.5

Move $7$ to the left of $x^{\frac{1}{3}}$ .

$f^{- 1} (\frac{x^{\frac{1}{3}}}{6} + 7) = x + 126 x^{\frac{2}{3}} + 5292 x^{\frac{1}{3}} + 74088 - 4536 (\frac{x^{\frac{2}{3}}}{36} + \frac{7 \cdot x^{\frac{1}{3}}}{6} + 7 (\frac{x^{\frac{1}{3}}}{6}) + 7 \cdot 7) + 31752 (\frac{x^{\frac{1}{3}}}{6} + 7) - 74088$

Step 5.2.3.7.1.6

Combine $7$ and $\frac{x^{\frac{1}{3}}}{6}$ .

$f^{- 1} (\frac{x^{\frac{1}{3}}}{6} + 7) = x + 126 x^{\frac{2}{3}} + 5292 x^{\frac{1}{3}} + 74088 - 4536 (\frac{x^{\frac{2}{3}}}{36} + \frac{7 x^{\frac{1}{3}}}{6} + \frac{7 x^{\frac{1}{3}}}{6} + 7 \cdot 7) + 31752 (\frac{x^{\frac{1}{3}}}{6} + 7) - 74088$

Step 5.2.3.7.1.7

Multiply $7$ by $7$ .

$f^{- 1} (\frac{x^{\frac{1}{3}}}{6} + 7) = x + 126 x^{\frac{2}{3}} + 5292 x^{\frac{1}{3}} + 74088 - 4536 (\frac{x^{\frac{2}{3}}}{36} + \frac{7 x^{\frac{1}{3}}}{6} + \frac{7 x^{\frac{1}{3}}}{6} + 49) + 31752 (\frac{x^{\frac{1}{3}}}{6} + 7) - 74088$

Step 5.2.3.7.2

Add $\frac{7 x^{\frac{1}{3}}}{6}$ and $\frac{7 x^{\frac{1}{3}}}{6}$ .

$f^{- 1} (\frac{x^{\frac{1}{3}}}{6} + 7) = x + 126 x^{\frac{2}{3}} + 5292 x^{\frac{1}{3}} + 74088 - 4536 (\frac{x^{\frac{2}{3}}}{36} + 2 (\frac{7 x^{\frac{1}{3}}}{6}) + 49) + 31752 (\frac{x^{\frac{1}{3}}}{6} + 7) - 74088$

Step 5.2.3.8

Cancel the common factor of $2$ .

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Step 5.2.3.8.1

Factor $2$ out of $6$ .

$f^{- 1} (\frac{x^{\frac{1}{3}}}{6} + 7) = x + 126 x^{\frac{2}{3}} + 5292 x^{\frac{1}{3}} + 74088 - 4536 (\frac{x^{\frac{2}{3}}}{36} + 2 (\frac{7 x^{\frac{1}{3}}}{2 (3)}) + 49) + 31752 (\frac{x^{\frac{1}{3}}}{6} + 7) - 74088$

Step 5.2.3.8.2

Cancel the common factor.

$f^{- 1} (\frac{x^{\frac{1}{3}}}{6} + 7) = x + 126 x^{\frac{2}{3}} + 5292 x^{\frac{1}{3}} + 74088 - 4536 (\frac{x^{\frac{2}{3}}}{36} + 2 (\frac{7 x^{\frac{1}{3}}}{2 \cdot 3}) + 49) + 31752 (\frac{x^{\frac{1}{3}}}{6} + 7) - 74088$

Step 5.2.3.8.3

Rewrite the expression.

$f^{- 1} (\frac{x^{\frac{1}{3}}}{6} + 7) = x + 126 x^{\frac{2}{3}} + 5292 x^{\frac{1}{3}} + 74088 - 4536 (\frac{x^{\frac{2}{3}}}{36} + \frac{7 x^{\frac{1}{3}}}{3} + 49) + 31752 (\frac{x^{\frac{1}{3}}}{6} + 7) - 74088$

Step 5.2.3.9

Apply the distributive property.

$f^{- 1} (\frac{x^{\frac{1}{3}}}{6} + 7) = x + 126 x^{\frac{2}{3}} + 5292 x^{\frac{1}{3}} + 74088 - 4536 \frac{x^{\frac{2}{3}}}{36} - 4536 \frac{7 x^{\frac{1}{3}}}{3} - 4536 \cdot 49 + 31752 (\frac{x^{\frac{1}{3}}}{6} + 7) - 74088$

Step 5.2.3.10

Simplify.

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Step 5.2.3.10.1

Cancel the common factor of $36$ .

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Step 5.2.3.10.1.1

Factor $36$ out of $- 4536$ .

$f^{- 1} (\frac{x^{\frac{1}{3}}}{6} + 7) = x + 126 x^{\frac{2}{3}} + 5292 x^{\frac{1}{3}} + 74088 + 36 (- 126) (\frac{x^{\frac{2}{3}}}{36}) - 4536 \frac{7 x^{\frac{1}{3}}}{3} - 4536 \cdot 49 + 31752 (\frac{x^{\frac{1}{3}}}{6} + 7) - 74088$

Step 5.2.3.10.1.2

Cancel the common factor.

$f^{- 1} (\frac{x^{\frac{1}{3}}}{6} + 7) = x + 126 x^{\frac{2}{3}} + 5292 x^{\frac{1}{3}} + 74088 + 36 \cdot (- 126 \frac{x^{\frac{2}{3}}}{36}) - 4536 \frac{7 x^{\frac{1}{3}}}{3} - 4536 \cdot 49 + 31752 (\frac{x^{\frac{1}{3}}}{6} + 7) - 74088$

Step 5.2.3.10.1.3

Rewrite the expression.

$f^{- 1} (\frac{x^{\frac{1}{3}}}{6} + 7) = x + 126 x^{\frac{2}{3}} + 5292 x^{\frac{1}{3}} + 74088 - 126 x^{\frac{2}{3}} - 4536 \frac{7 x^{\frac{1}{3}}}{3} - 4536 \cdot 49 + 31752 (\frac{x^{\frac{1}{3}}}{6} + 7) - 74088$

Step 5.2.3.10.2

Cancel the common factor of $3$ .

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Step 5.2.3.10.2.1

Factor $3$ out of $- 4536$ .

$f^{- 1} (\frac{x^{\frac{1}{3}}}{6} + 7) = x + 126 x^{\frac{2}{3}} + 5292 x^{\frac{1}{3}} + 74088 - 126 x^{\frac{2}{3}} + 3 (- 1512) (\frac{7 x^{\frac{1}{3}}}{3}) - 4536 \cdot 49 + 31752 (\frac{x^{\frac{1}{3}}}{6} + 7) - 74088$

Step 5.2.3.10.2.2

Cancel the common factor.

$f^{- 1} (\frac{x^{\frac{1}{3}}}{6} + 7) = x + 126 x^{\frac{2}{3}} + 5292 x^{\frac{1}{3}} + 74088 - 126 x^{\frac{2}{3}} + 3 \cdot (- 1512 \frac{7 x^{\frac{1}{3}}}{3}) - 4536 \cdot 49 + 31752 (\frac{x^{\frac{1}{3}}}{6} + 7) - 74088$

Step 5.2.3.10.2.3

Rewrite the expression.

$f^{- 1} (\frac{x^{\frac{1}{3}}}{6} + 7) = x + 126 x^{\frac{2}{3}} + 5292 x^{\frac{1}{3}} + 74088 - 126 x^{\frac{2}{3}} - 1512 (7 x^{\frac{1}{3}}) - 4536 \cdot 49 + 31752 (\frac{x^{\frac{1}{3}}}{6} + 7) - 74088$

Step 5.2.3.10.3

Multiply $7$ by $- 1512$ .

$f^{- 1} (\frac{x^{\frac{1}{3}}}{6} + 7) = x + 126 x^{\frac{2}{3}} + 5292 x^{\frac{1}{3}} + 74088 - 126 x^{\frac{2}{3}} - 10584 x^{\frac{1}{3}} - 4536 \cdot 49 + 31752 (\frac{x^{\frac{1}{3}}}{6} + 7) - 74088$

Step 5.2.3.10.4

Multiply $- 4536$ by $49$ .

$f^{- 1} (\frac{x^{\frac{1}{3}}}{6} + 7) = x + 126 x^{\frac{2}{3}} + 5292 x^{\frac{1}{3}} + 74088 - 126 x^{\frac{2}{3}} - 10584 x^{\frac{1}{3}} - 222264 + 31752 (\frac{x^{\frac{1}{3}}}{6} + 7) - 74088$

Step 5.2.3.11

Apply the distributive property.

Step 5.2.3.12

Cancel the common factor of $6$ .

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Step 5.2.3.12.1

Factor $6$ out of $31752$ .

$f^{- 1} (\frac{x^{\frac{1}{3}}}{6} + 7) = x + 126 x^{\frac{2}{3}} + 5292 x^{\frac{1}{3}} + 74088 - 126 x^{\frac{2}{3}} - 10584 x^{\frac{1}{3}} - 222264 + 6 (5292) (\frac{x^{\frac{1}{3}}}{6}) + 31752 \cdot 7 - 74088$

Step 5.2.3.12.2

Cancel the common factor.

$f^{- 1} (\frac{x^{\frac{1}{3}}}{6} + 7) = x + 126 x^{\frac{2}{3}} + 5292 x^{\frac{1}{3}} + 74088 - 126 x^{\frac{2}{3}} - 10584 x^{\frac{1}{3}} - 222264 + 6 \cdot (5292 (\frac{x^{\frac{1}{3}}}{6})) + 31752 \cdot 7 - 74088$

Step 5.2.3.12.3

Rewrite the expression.

$f^{- 1} (\frac{x^{\frac{1}{3}}}{6} + 7) = x + 126 x^{\frac{2}{3}} + 5292 x^{\frac{1}{3}} + 74088 - 126 x^{\frac{2}{3}} - 10584 x^{\frac{1}{3}} - 222264 + 5292 x^{\frac{1}{3}} + 31752 \cdot 7 - 74088$

Step 5.2.3.13

Multiply $31752$ by $7$ .

$f^{- 1} (\frac{x^{\frac{1}{3}}}{6} + 7) = x + 126 x^{\frac{2}{3}} + 5292 x^{\frac{1}{3}} + 74088 - 126 x^{\frac{2}{3}} - 10584 x^{\frac{1}{3}} - 222264 + 5292 x^{\frac{1}{3}} + 222264 - 74088$

Step 5.2.4

Simplify by adding terms.

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Step 5.2.4.1

Combine the opposite terms in $x + 126 x^{\frac{2}{3}} + 5292 x^{\frac{1}{3}} + 74088 - 126 x^{\frac{2}{3}} - 10584 x^{\frac{1}{3}} - 222264 + 5292 x^{\frac{1}{3}} + 222264 - 74088$ .

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Step 5.2.4.1.1

Subtract $126 x^{\frac{2}{3}}$ from $126 x^{\frac{2}{3}}$ .

$f^{- 1} (\frac{x^{\frac{1}{3}}}{6} + 7) = x + 5292 x^{\frac{1}{3}} + 74088 + 0 - 10584 x^{\frac{1}{3}} - 222264 + 5292 x^{\frac{1}{3}} + 222264 - 74088$

Step 5.2.4.1.2

Add $x + 5292 x^{\frac{1}{3}} + 74088$ and $0$ .

$f^{- 1} (\frac{x^{\frac{1}{3}}}{6} + 7) = x + 5292 x^{\frac{1}{3}} + 74088 - 10584 x^{\frac{1}{3}} - 222264 + 5292 x^{\frac{1}{3}} + 222264 - 74088$

Step 5.2.4.1.3

Add $- 222264$ and $222264$ .

$f^{- 1} (\frac{x^{\frac{1}{3}}}{6} + 7) = x + 5292 x^{\frac{1}{3}} + 74088 - 10584 x^{\frac{1}{3}} + 5292 x^{\frac{1}{3}} + 0 - 74088$

Step 5.2.4.1.4

Add $x + 5292 x^{\frac{1}{3}} + 74088 - 10584 x^{\frac{1}{3}} + 5292 x^{\frac{1}{3}}$ and $0$ .

$f^{- 1} (\frac{x^{\frac{1}{3}}}{6} + 7) = x + 5292 x^{\frac{1}{3}} + 74088 - 10584 x^{\frac{1}{3}} + 5292 x^{\frac{1}{3}} - 74088$

Step 5.2.4.1.5

Subtract $74088$ from $74088$ .

$f^{- 1} (\frac{x^{\frac{1}{3}}}{6} + 7) = x + 5292 x^{\frac{1}{3}} - 10584 x^{\frac{1}{3}} + 5292 x^{\frac{1}{3}} + 0$

Step 5.2.4.1.6

Add $x + 5292 x^{\frac{1}{3}} - 10584 x^{\frac{1}{3}} + 5292 x^{\frac{1}{3}}$ and $0$ .

$f^{- 1} (\frac{x^{\frac{1}{3}}}{6} + 7) = x + 5292 x^{\frac{1}{3}} - 10584 x^{\frac{1}{3}} + 5292 x^{\frac{1}{3}}$

Step 5.2.4.2

Subtract $10584 x^{\frac{1}{3}}$ from $5292 x^{\frac{1}{3}}$ .

$f^{- 1} (\frac{x^{\frac{1}{3}}}{6} + 7) = x - 5292 x^{\frac{1}{3}} + 5292 x^{\frac{1}{3}}$

Step 5.2.4.3

Combine the opposite terms in $x - 5292 x^{\frac{1}{3}} + 5292 x^{\frac{1}{3}}$ .

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Step 5.2.4.3.1

Add $- 5292 x^{\frac{1}{3}}$ and $5292 x^{\frac{1}{3}}$ .

$f^{- 1} (\frac{x^{\frac{1}{3}}}{6} + 7) = x + 0$

Step 5.2.4.3.2

Add $x$ and $0$ .

$f^{- 1} (\frac{x^{\frac{1}{3}}}{6} + 7) = x$

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 $f (216 x^{3} - 4536 x^{2} + 31752 x - 74088)$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (216 x^{3} - 4536 x^{2} + 31752 x - 74088) = \frac{{(216 x^{3} - 4536 x^{2} + 31752 x - 74088)}^{\frac{1}{3}}}{6} + 7$

Step 5.3.3

To write $7$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$f (216 x^{3} - 4536 x^{2} + 31752 x - 74088) = \frac{{(216 x^{3} - 4536 x^{2} + 31752 x - 74088)}^{\frac{1}{3}}}{6} + 7 \cdot \frac{6}{6}$

Step 5.3.4

Combine $7$ and $\frac{6}{6}$ .

$f (216 x^{3} - 4536 x^{2} + 31752 x - 74088) = \frac{{(216 x^{3} - 4536 x^{2} + 31752 x - 74088)}^{\frac{1}{3}}}{6} + \frac{7 \cdot 6}{6}$

Step 5.3.5

Combine the numerators over the common denominator.

$f (216 x^{3} - 4536 x^{2} + 31752 x - 74088) = \frac{{(216 x^{3} - 4536 x^{2} + 31752 x - 74088)}^{\frac{1}{3}} + 7 \cdot 6}{6}$

Step 5.3.6

Multiply $7$ by $6$ .

$f (216 x^{3} - 4536 x^{2} + 31752 x - 74088) = \frac{{(216 x^{3} - 4536 x^{2} + 31752 x - 74088)}^{\frac{1}{3}} + 42}{6}$

Step 5.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = 216 x^{3} - 4536 x^{2} + 31752 x - 74088$ is the inverse of $f (x) = \frac{x^{\frac{1}{3}}}{6} + 7$ .

$f^{- 1} (x) = 216 x^{3} - 4536 x^{2} + 31752 x - 74088$