Algebra Examples

$f (x) = 2 {(x - 6)}^{\frac{1}{3}} + 10$

Step 1

Write $f (x) = 2 {(x - 6)}^{\frac{1}{3}} + 10$ as an equation.

$y = 2 {(x - 6)}^{\frac{1}{3}} + 10$

Step 2

Interchange the variables.

$x = 2 {(y - 6)}^{\frac{1}{3}} + 10$

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $2 {(y - 6)}^{\frac{1}{3}} + 10 = x$ .

$2 {(y - 6)}^{\frac{1}{3}} + 10 = x$

Step 3.2

Subtract $10$ from both sides of the equation.

$2 {(y - 6)}^{\frac{1}{3}} = x - 10$

Step 3.3

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

${(2 {(y - 6)}^{\frac{1}{3}})}^{3} = {(x - 10)}^{3}$

Step 3.4

Simplify the exponent.

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Step 3.4.1

Simplify the left side.

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Step 3.4.1.1

Simplify ${(2 {(y - 6)}^{\frac{1}{3}})}^{3}$ .

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Step 3.4.1.1.1

Apply the product rule to $2 {(y - 6)}^{\frac{1}{3}}$ .

$2^{3} {({(y - 6)}^{\frac{1}{3}})}^{3} = {(x - 10)}^{3}$

Step 3.4.1.1.2

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

$8 {({(y - 6)}^{\frac{1}{3}})}^{3} = {(x - 10)}^{3}$

Step 3.4.1.1.3

Multiply the exponents in ${({(y - 6)}^{\frac{1}{3}})}^{3}$ .

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Step 3.4.1.1.3.1

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

$8 {(y - 6)}^{\frac{1}{3} \cdot 3} = {(x - 10)}^{3}$

Step 3.4.1.1.3.2

Cancel the common factor of $3$ .

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Step 3.4.1.1.3.2.1

Cancel the common factor.

$8 {(y - 6)}^{\frac{1}{3} \cdot 3} = {(x - 10)}^{3}$

Step 3.4.1.1.3.2.2

Rewrite the expression.

$8 {(y - 6)}^{1} = {(x - 10)}^{3}$

Step 3.4.1.1.4

Simplify.

$8 (y - 6) = {(x - 10)}^{3}$

Step 3.4.1.1.5

Apply the distributive property.

$8 y + 8 \cdot - 6 = {(x - 10)}^{3}$

Step 3.4.1.1.6

Multiply $8$ by $- 6$ .

$8 y - 48 = {(x - 10)}^{3}$

Step 3.4.2

Simplify the right side.

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Step 3.4.2.1

Simplify ${(x - 10)}^{3}$ .

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Step 3.4.2.1.1

Use the Binomial Theorem.

$8 y - 48 = x^{3} + 3 x^{2} \cdot - 10 + 3 x {(- 10)}^{2} + {(- 10)}^{3}$

Step 3.4.2.1.2

Simplify each term.

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Step 3.4.2.1.2.1

Multiply $- 10$ by $3$ .

$8 y - 48 = x^{3} - 30 x^{2} + 3 x {(- 10)}^{2} + {(- 10)}^{3}$

Step 3.4.2.1.2.2

Raise $- 10$ to the power of $2$ .

$8 y - 48 = x^{3} - 30 x^{2} + 3 x \cdot 100 + {(- 10)}^{3}$

Step 3.4.2.1.2.3

Multiply $100$ by $3$ .

$8 y - 48 = x^{3} - 30 x^{2} + 300 x + {(- 10)}^{3}$

Step 3.4.2.1.2.4

Raise $- 10$ to the power of $3$ .

$8 y - 48 = x^{3} - 30 x^{2} + 300 x - 1000$

Step 3.5

Solve for $y$ .

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Step 3.5.1

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

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Step 3.5.1.1

Add $48$ to both sides of the equation.

$8 y = x^{3} - 30 x^{2} + 300 x - 1000 + 48$

Step 3.5.1.2

Add $- 1000$ and $48$ .

$8 y = x^{3} - 30 x^{2} + 300 x - 952$

Step 3.5.2

Divide each term in $8 y = x^{3} - 30 x^{2} + 300 x - 952$ by $8$ and simplify.

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Step 3.5.2.1

Divide each term in $8 y = x^{3} - 30 x^{2} + 300 x - 952$ by $8$ .

$\frac{8 y}{8} = \frac{x^{3}}{8} + \frac{- 30 x^{2}}{8} + \frac{300 x}{8} + \frac{- 952}{8}$

Step 3.5.2.2

Simplify the left side.

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Step 3.5.2.2.1

Cancel the common factor of $8$ .

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Step 3.5.2.2.1.1

Cancel the common factor.

$\frac{8 y}{8} = \frac{x^{3}}{8} + \frac{- 30 x^{2}}{8} + \frac{300 x}{8} + \frac{- 952}{8}$

Step 3.5.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{x^{3}}{8} + \frac{- 30 x^{2}}{8} + \frac{300 x}{8} + \frac{- 952}{8}$

Step 3.5.2.3

Simplify the right side.

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Step 3.5.2.3.1

Simplify each term.

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Step 3.5.2.3.1.1

Cancel the common factor of $- 30$ and $8$ .

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Step 3.5.2.3.1.1.1

Factor $2$ out of $- 30 x^{2}$ .

$y = \frac{x^{3}}{8} + \frac{2 (- 15 x^{2})}{8} + \frac{300 x}{8} + \frac{- 952}{8}$

Step 3.5.2.3.1.1.2

Cancel the common factors.

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Step 3.5.2.3.1.1.2.1

Factor $2$ out of $8$ .

$y = \frac{x^{3}}{8} + \frac{2 (- 15 x^{2})}{2 (4)} + \frac{300 x}{8} + \frac{- 952}{8}$

Step 3.5.2.3.1.1.2.2

Cancel the common factor.

$y = \frac{x^{3}}{8} + \frac{2 (- 15 x^{2})}{2 \cdot 4} + \frac{300 x}{8} + \frac{- 952}{8}$

Step 3.5.2.3.1.1.2.3

Rewrite the expression.

$y = \frac{x^{3}}{8} + \frac{- 15 x^{2}}{4} + \frac{300 x}{8} + \frac{- 952}{8}$

Step 3.5.2.3.1.2

Move the negative in front of the fraction.

$y = \frac{x^{3}}{8} - \frac{15 x^{2}}{4} + \frac{300 x}{8} + \frac{- 952}{8}$

Step 3.5.2.3.1.3

Cancel the common factor of $300$ and $8$ .

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Step 3.5.2.3.1.3.1

Factor $4$ out of $300 x$ .

$y = \frac{x^{3}}{8} - \frac{15 x^{2}}{4} + \frac{4 (75 x)}{8} + \frac{- 952}{8}$

Step 3.5.2.3.1.3.2

Cancel the common factors.

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Step 3.5.2.3.1.3.2.1

Factor $4$ out of $8$ .

$y = \frac{x^{3}}{8} - \frac{15 x^{2}}{4} + \frac{4 (75 x)}{4 (2)} + \frac{- 952}{8}$

Step 3.5.2.3.1.3.2.2

Cancel the common factor.

$y = \frac{x^{3}}{8} - \frac{15 x^{2}}{4} + \frac{4 (75 x)}{4 \cdot 2} + \frac{- 952}{8}$

Step 3.5.2.3.1.3.2.3

Rewrite the expression.

$y = \frac{x^{3}}{8} - \frac{15 x^{2}}{4} + \frac{75 x}{2} + \frac{- 952}{8}$

Step 3.5.2.3.1.4

Divide $- 952$ by $8$ .

$y = \frac{x^{3}}{8} - \frac{15 x^{2}}{4} + \frac{75 x}{2} - 119$

Step 4

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

$f^{- 1} (x) = \frac{x^{3}}{8} - \frac{15 x^{2}}{4} + \frac{75 x}{2} - 119$

Step 5

Verify if $f^{- 1} (x) = \frac{x^{3}}{8} - \frac{15 x^{2}}{4} + \frac{75 x}{2} - 119$ is the inverse of $f (x) = 2 {(x - 6)}^{\frac{1}{3}} + 10$ .

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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 $f^{- 1} (2 {(x - 6)}^{\frac{1}{3}} + 10)$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (2 {(x - 6)}^{\frac{1}{3}} + 10) = \frac{{(2 {(x - 6)}^{\frac{1}{3}} + 10)}^{3}}{8} - \frac{15 {(2 {(x - 6)}^{\frac{1}{3}} + 10)}^{2}}{4} + \frac{75 (2 {(x - 6)}^{\frac{1}{3}} + 10)}{2} - 119$

Step 5.2.3

Simplify each term.

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Step 5.2.3.1

Simplify the numerator.

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Step 5.2.3.1.1

Factor $2$ out of $2 {(x - 6)}^{\frac{1}{3}} + 10$ .

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Step 5.2.3.1.1.1

Factor $2$ out of $2 {(x - 6)}^{\frac{1}{3}}$ .

$f^{- 1} (2 {(x - 6)}^{\frac{1}{3}} + 10) = \frac{{(2 ({(x - 6)}^{\frac{1}{3}}) + 10)}^{3}}{8} - \frac{15 {(2 {(x - 6)}^{\frac{1}{3}} + 10)}^{2}}{4} + \frac{75 (2 {(x - 6)}^{\frac{1}{3}} + 10)}{2} - 119$

Step 5.2.3.1.1.2

Factor $2$ out of $10$ .

$f^{- 1} (2 {(x - 6)}^{\frac{1}{3}} + 10) = \frac{{(2 {(x - 6)}^{\frac{1}{3}} + 2 \cdot 5)}^{3}}{8} - \frac{15 {(2 {(x - 6)}^{\frac{1}{3}} + 10)}^{2}}{4} + \frac{75 (2 {(x - 6)}^{\frac{1}{3}} + 10)}{2} - 119$

Step 5.2.3.1.1.3

Factor $2$ out of $2 {(x - 6)}^{\frac{1}{3}} + 2 \cdot 5$ .

$f^{- 1} (2 {(x - 6)}^{\frac{1}{3}} + 10) = \frac{{(2 ({(x - 6)}^{\frac{1}{3}} + 5))}^{3}}{8} - \frac{15 {(2 {(x - 6)}^{\frac{1}{3}} + 10)}^{2}}{4} + \frac{75 (2 {(x - 6)}^{\frac{1}{3}} + 10)}{2} - 119$

Step 5.2.3.1.2

Apply the product rule to $2 ({(x - 6)}^{\frac{1}{3}} + 5)$ .

$f^{- 1} (2 {(x - 6)}^{\frac{1}{3}} + 10) = \frac{2^{3} {({(x - 6)}^{\frac{1}{3}} + 5)}^{3}}{8} - \frac{15 {(2 {(x - 6)}^{\frac{1}{3}} + 10)}^{2}}{4} + \frac{75 (2 {(x - 6)}^{\frac{1}{3}} + 10)}{2} - 119$

Step 5.2.3.1.3

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

$f^{- 1} (2 {(x - 6)}^{\frac{1}{3}} + 10) = \frac{8 {({(x - 6)}^{\frac{1}{3}} + 5)}^{3}}{8} - \frac{15 {(2 {(x - 6)}^{\frac{1}{3}} + 10)}^{2}}{4} + \frac{75 (2 {(x - 6)}^{\frac{1}{3}} + 10)}{2} - 119$

Step 5.2.3.2

Cancel the common factor.

Step 5.2.3.3

Divide ${({(x - 6)}^{\frac{1}{3}} + 5)}^{3}$ by $1$ .

$f^{- 1} (2 {(x - 6)}^{\frac{1}{3}} + 10) = {({(x - 6)}^{\frac{1}{3}} + 5)}^{3} - \frac{15 {(2 {(x - 6)}^{\frac{1}{3}} + 10)}^{2}}{4} + \frac{75 (2 {(x - 6)}^{\frac{1}{3}} + 10)}{2} - 119$

Step 5.2.3.4

Use the Binomial Theorem.

$f^{- 1} (2 {(x - 6)}^{\frac{1}{3}} + 10) = {({(x - 6)}^{\frac{1}{3}})}^{3} + 3 {({(x - 6)}^{\frac{1}{3}})}^{2} \cdot 5 + 3 {(x - 6)}^{\frac{1}{3}} \cdot 5^{2} + 5^{3} - \frac{15 {(2 {(x - 6)}^{\frac{1}{3}} + 10)}^{2}}{4} + \frac{75 (2 {(x - 6)}^{\frac{1}{3}} + 10)}{2} - 119$

Step 5.2.3.5

Simplify each term.

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Step 5.2.3.5.1

Multiply the exponents in ${({(x - 6)}^{\frac{1}{3}})}^{3}$ .

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Step 5.2.3.5.1.1

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

$f^{- 1} (2 {(x - 6)}^{\frac{1}{3}} + 10) = {(x - 6)}^{\frac{1}{3} \cdot 3} + 3 {({(x - 6)}^{\frac{1}{3}})}^{2} \cdot 5 + 3 {(x - 6)}^{\frac{1}{3}} \cdot 5^{2} + 5^{3} - \frac{15 {(2 {(x - 6)}^{\frac{1}{3}} + 10)}^{2}}{4} + \frac{75 (2 {(x - 6)}^{\frac{1}{3}} + 10)}{2} - 119$

Step 5.2.3.5.1.2

Cancel the common factor of $3$ .

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Step 5.2.3.5.1.2.1

Cancel the common factor.

Step 5.2.3.5.1.2.2

Rewrite the expression.

$f^{- 1} (2 {(x - 6)}^{\frac{1}{3}} + 10) = (x - 6) + 3 {({(x - 6)}^{\frac{1}{3}})}^{2} \cdot 5 + 3 {(x - 6)}^{\frac{1}{3}} \cdot 5^{2} + 5^{3} - \frac{15 {(2 {(x - 6)}^{\frac{1}{3}} + 10)}^{2}}{4} + \frac{75 (2 {(x - 6)}^{\frac{1}{3}} + 10)}{2} - 119$

Step 5.2.3.5.2

Simplify.

$f^{- 1} (2 {(x - 6)}^{\frac{1}{3}} + 10) = x - 6 + 3 {({(x - 6)}^{\frac{1}{3}})}^{2} \cdot 5 + 3 {(x - 6)}^{\frac{1}{3}} \cdot 5^{2} + 5^{3} - \frac{15 {(2 {(x - 6)}^{\frac{1}{3}} + 10)}^{2}}{4} + \frac{75 (2 {(x - 6)}^{\frac{1}{3}} + 10)}{2} - 119$

Step 5.2.3.5.3

Multiply the exponents in ${({(x - 6)}^{\frac{1}{3}})}^{2}$ .

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Step 5.2.3.5.3.1

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

$f^{- 1} (2 {(x - 6)}^{\frac{1}{3}} + 10) = x - 6 + 3 {(x - 6)}^{\frac{1}{3} \cdot 2} \cdot 5 + 3 {(x - 6)}^{\frac{1}{3}} \cdot 5^{2} + 5^{3} - \frac{15 {(2 {(x - 6)}^{\frac{1}{3}} + 10)}^{2}}{4} + \frac{75 (2 {(x - 6)}^{\frac{1}{3}} + 10)}{2} - 119$

Step 5.2.3.5.3.2

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

$f^{- 1} (2 {(x - 6)}^{\frac{1}{3}} + 10) = x - 6 + 3 {(x - 6)}^{\frac{2}{3}} \cdot 5 + 3 {(x - 6)}^{\frac{1}{3}} \cdot 5^{2} + 5^{3} - \frac{15 {(2 {(x - 6)}^{\frac{1}{3}} + 10)}^{2}}{4} + \frac{75 (2 {(x - 6)}^{\frac{1}{3}} + 10)}{2} - 119$

Step 5.2.3.5.4

Multiply $5$ by $3$ .

$f^{- 1} (2 {(x - 6)}^{\frac{1}{3}} + 10) = x - 6 + 15 {(x - 6)}^{\frac{2}{3}} + 3 {(x - 6)}^{\frac{1}{3}} \cdot 5^{2} + 5^{3} - \frac{15 {(2 {(x - 6)}^{\frac{1}{3}} + 10)}^{2}}{4} + \frac{75 (2 {(x - 6)}^{\frac{1}{3}} + 10)}{2} - 119$

Step 5.2.3.5.5

Raise $5$ to the power of $2$ .

$f^{- 1} (2 {(x - 6)}^{\frac{1}{3}} + 10) = x - 6 + 15 {(x - 6)}^{\frac{2}{3}} + 3 {(x - 6)}^{\frac{1}{3}} \cdot 25 + 5^{3} - \frac{15 {(2 {(x - 6)}^{\frac{1}{3}} + 10)}^{2}}{4} + \frac{75 (2 {(x - 6)}^{\frac{1}{3}} + 10)}{2} - 119$

Step 5.2.3.5.6

Multiply $25$ by $3$ .

$f^{- 1} (2 {(x - 6)}^{\frac{1}{3}} + 10) = x - 6 + 15 {(x - 6)}^{\frac{2}{3}} + 75 {(x - 6)}^{\frac{1}{3}} + 5^{3} - \frac{15 {(2 {(x - 6)}^{\frac{1}{3}} + 10)}^{2}}{4} + \frac{75 (2 {(x - 6)}^{\frac{1}{3}} + 10)}{2} - 119$

Step 5.2.3.5.7

Raise $5$ to the power of $3$ .

$f^{- 1} (2 {(x - 6)}^{\frac{1}{3}} + 10) = x - 6 + 15 {(x - 6)}^{\frac{2}{3}} + 75 {(x - 6)}^{\frac{1}{3}} + 125 - \frac{15 {(2 {(x - 6)}^{\frac{1}{3}} + 10)}^{2}}{4} + \frac{75 (2 {(x - 6)}^{\frac{1}{3}} + 10)}{2} - 119$

Step 5.2.3.6

Add $- 6$ and $125$ .

$f^{- 1} (2 {(x - 6)}^{\frac{1}{3}} + 10) = x + 15 {(x - 6)}^{\frac{2}{3}} + 75 {(x - 6)}^{\frac{1}{3}} + 119 - \frac{15 {(2 {(x - 6)}^{\frac{1}{3}} + 10)}^{2}}{4} + \frac{75 (2 {(x - 6)}^{\frac{1}{3}} + 10)}{2} - 119$

Step 5.2.3.7

Simplify the numerator.

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Step 5.2.3.7.1

Factor $2$ out of $2 {(x - 6)}^{\frac{1}{3}} + 10$ .

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Step 5.2.3.7.1.1

Factor $2$ out of $2 {(x - 6)}^{\frac{1}{3}}$ .

$f^{- 1} (2 {(x - 6)}^{\frac{1}{3}} + 10) = x + 15 {(x - 6)}^{\frac{2}{3}} + 75 {(x - 6)}^{\frac{1}{3}} + 119 - \frac{15 {(2 ({(x - 6)}^{\frac{1}{3}}) + 10)}^{2}}{4} + \frac{75 (2 {(x - 6)}^{\frac{1}{3}} + 10)}{2} - 119$

Step 5.2.3.7.1.2

Factor $2$ out of $10$ .

$f^{- 1} (2 {(x - 6)}^{\frac{1}{3}} + 10) = x + 15 {(x - 6)}^{\frac{2}{3}} + 75 {(x - 6)}^{\frac{1}{3}} + 119 - \frac{15 {(2 {(x - 6)}^{\frac{1}{3}} + 2 \cdot 5)}^{2}}{4} + \frac{75 (2 {(x - 6)}^{\frac{1}{3}} + 10)}{2} - 119$

Step 5.2.3.7.1.3

Factor $2$ out of $2 {(x - 6)}^{\frac{1}{3}} + 2 \cdot 5$ .

$f^{- 1} (2 {(x - 6)}^{\frac{1}{3}} + 10) = x + 15 {(x - 6)}^{\frac{2}{3}} + 75 {(x - 6)}^{\frac{1}{3}} + 119 - \frac{15 {(2 ({(x - 6)}^{\frac{1}{3}} + 5))}^{2}}{4} + \frac{75 (2 {(x - 6)}^{\frac{1}{3}} + 10)}{2} - 119$

Step 5.2.3.7.2

Apply the product rule to $2 ({(x - 6)}^{\frac{1}{3}} + 5)$ .

$f^{- 1} (2 {(x - 6)}^{\frac{1}{3}} + 10) = x + 15 {(x - 6)}^{\frac{2}{3}} + 75 {(x - 6)}^{\frac{1}{3}} + 119 - \frac{15 \cdot (2^{2} {({(x - 6)}^{\frac{1}{3}} + 5)}^{2})}{4} + \frac{75 (2 {(x - 6)}^{\frac{1}{3}} + 10)}{2} - 119$

Step 5.2.3.7.3

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

$f^{- 1} (2 {(x - 6)}^{\frac{1}{3}} + 10) = x + 15 {(x - 6)}^{\frac{2}{3}} + 75 {(x - 6)}^{\frac{1}{3}} + 119 - \frac{15 \cdot (4 {({(x - 6)}^{\frac{1}{3}} + 5)}^{2})}{4} + \frac{75 (2 {(x - 6)}^{\frac{1}{3}} + 10)}{2} - 119$

Step 5.2.3.7.4

Multiply $15$ by $4$ .

$f^{- 1} (2 {(x - 6)}^{\frac{1}{3}} + 10) = x + 15 {(x - 6)}^{\frac{2}{3}} + 75 {(x - 6)}^{\frac{1}{3}} + 119 - \frac{60 {({(x - 6)}^{\frac{1}{3}} + 5)}^{2}}{4} + \frac{75 (2 {(x - 6)}^{\frac{1}{3}} + 10)}{2} - 119$

Step 5.2.3.8

Factor $4$ out of $60 {({(x - 6)}^{\frac{1}{3}} + 5)}^{2}$ .

$f^{- 1} (2 {(x - 6)}^{\frac{1}{3}} + 10) = x + 15 {(x - 6)}^{\frac{2}{3}} + 75 {(x - 6)}^{\frac{1}{3}} + 119 - \frac{4 (15 {({(x - 6)}^{\frac{1}{3}} + 5)}^{2})}{4} + \frac{75 (2 {(x - 6)}^{\frac{1}{3}} + 10)}{2} - 119$

Step 5.2.3.9

Cancel the common factors.

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Step 5.2.3.9.1

Factor $4$ out of $4$ .

$f^{- 1} (2 {(x - 6)}^{\frac{1}{3}} + 10) = x + 15 {(x - 6)}^{\frac{2}{3}} + 75 {(x - 6)}^{\frac{1}{3}} + 119 - \frac{4 (15 {({(x - 6)}^{\frac{1}{3}} + 5)}^{2})}{4 (1)} + \frac{75 (2 {(x - 6)}^{\frac{1}{3}} + 10)}{2} - 119$

Step 5.2.3.9.2

Cancel the common factor.

$f^{- 1} (2 {(x - 6)}^{\frac{1}{3}} + 10) = x + 15 {(x - 6)}^{\frac{2}{3}} + 75 {(x - 6)}^{\frac{1}{3}} + 119 - \frac{4 (15 {({(x - 6)}^{\frac{1}{3}} + 5)}^{2})}{4 \cdot 1} + \frac{75 (2 {(x - 6)}^{\frac{1}{3}} + 10)}{2} - 119$

Step 5.2.3.9.3

Rewrite the expression.

$f^{- 1} (2 {(x - 6)}^{\frac{1}{3}} + 10) = x + 15 {(x - 6)}^{\frac{2}{3}} + 75 {(x - 6)}^{\frac{1}{3}} + 119 - \frac{15 {({(x - 6)}^{\frac{1}{3}} + 5)}^{2}}{1} + \frac{75 (2 {(x - 6)}^{\frac{1}{3}} + 10)}{2} - 119$

Step 5.2.3.9.4

Divide $15 {({(x - 6)}^{\frac{1}{3}} + 5)}^{2}$ by $1$ .

$f^{- 1} (2 {(x - 6)}^{\frac{1}{3}} + 10) = x + 15 {(x - 6)}^{\frac{2}{3}} + 75 {(x - 6)}^{\frac{1}{3}} + 119 - (15 {({(x - 6)}^{\frac{1}{3}} + 5)}^{2}) + \frac{75 (2 {(x - 6)}^{\frac{1}{3}} + 10)}{2} - 119$

Step 5.2.3.10

Rewrite ${({(x - 6)}^{\frac{1}{3}} + 5)}^{2}$ as $({(x - 6)}^{\frac{1}{3}} + 5) ({(x - 6)}^{\frac{1}{3}} + 5)$ .

$f^{- 1} (2 {(x - 6)}^{\frac{1}{3}} + 10) = x + 15 {(x - 6)}^{\frac{2}{3}} + 75 {(x - 6)}^{\frac{1}{3}} + 119 - (15 (({(x - 6)}^{\frac{1}{3}} + 5) ({(x - 6)}^{\frac{1}{3}} + 5))) + \frac{75 (2 {(x - 6)}^{\frac{1}{3}} + 10)}{2} - 119$

Step 5.2.3.11

Expand $({(x - 6)}^{\frac{1}{3}} + 5) ({(x - 6)}^{\frac{1}{3}} + 5)$ using the FOIL Method.

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Step 5.2.3.11.1

Apply the distributive property.

$f^{- 1} (2 {(x - 6)}^{\frac{1}{3}} + 10) = x + 15 {(x - 6)}^{\frac{2}{3}} + 75 {(x - 6)}^{\frac{1}{3}} + 119 - (15 ({(x - 6)}^{\frac{1}{3}} ({(x - 6)}^{\frac{1}{3}} + 5) + 5 ({(x - 6)}^{\frac{1}{3}} + 5))) + \frac{75 (2 {(x - 6)}^{\frac{1}{3}} + 10)}{2} - 119$

Step 5.2.3.11.2

Apply the distributive property.

$f^{- 1} (2 {(x - 6)}^{\frac{1}{3}} + 10) = x + 15 {(x - 6)}^{\frac{2}{3}} + 75 {(x - 6)}^{\frac{1}{3}} + 119 - (15 ({(x - 6)}^{\frac{1}{3}} {(x - 6)}^{\frac{1}{3}} + {(x - 6)}^{\frac{1}{3}} \cdot 5 + 5 ({(x - 6)}^{\frac{1}{3}} + 5))) + \frac{75 (2 {(x - 6)}^{\frac{1}{3}} + 10)}{2} - 119$

Step 5.2.3.11.3

Apply the distributive property.

$f^{- 1} (2 {(x - 6)}^{\frac{1}{3}} + 10) = x + 15 {(x - 6)}^{\frac{2}{3}} + 75 {(x - 6)}^{\frac{1}{3}} + 119 - (15 ({(x - 6)}^{\frac{1}{3}} {(x - 6)}^{\frac{1}{3}} + {(x - 6)}^{\frac{1}{3}} \cdot 5 + 5 {(x - 6)}^{\frac{1}{3}} + 5 \cdot 5)) + \frac{75 (2 {(x - 6)}^{\frac{1}{3}} + 10)}{2} - 119$

Step 5.2.3.12

Simplify and combine like terms.

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Step 5.2.3.12.1

Simplify each term.

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Step 5.2.3.12.1.1

Multiply ${(x - 6)}^{\frac{1}{3}}$ by ${(x - 6)}^{\frac{1}{3}}$ by adding the exponents.

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Step 5.2.3.12.1.1.1

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

$f^{- 1} (2 {(x - 6)}^{\frac{1}{3}} + 10) = x + 15 {(x - 6)}^{\frac{2}{3}} + 75 {(x - 6)}^{\frac{1}{3}} + 119 - (15 ({(x - 6)}^{\frac{1}{3} + \frac{1}{3}} + {(x - 6)}^{\frac{1}{3}} \cdot 5 + 5 {(x - 6)}^{\frac{1}{3}} + 5 \cdot 5)) + \frac{75 (2 {(x - 6)}^{\frac{1}{3}} + 10)}{2} - 119$

Step 5.2.3.12.1.1.2

Combine the numerators over the common denominator.

$f^{- 1} (2 {(x - 6)}^{\frac{1}{3}} + 10) = x + 15 {(x - 6)}^{\frac{2}{3}} + 75 {(x - 6)}^{\frac{1}{3}} + 119 - (15 ({(x - 6)}^{\frac{1 + 1}{3}} + {(x - 6)}^{\frac{1}{3}} \cdot 5 + 5 {(x - 6)}^{\frac{1}{3}} + 5 \cdot 5)) + \frac{75 (2 {(x - 6)}^{\frac{1}{3}} + 10)}{2} - 119$

Step 5.2.3.12.1.1.3

Add $1$ and $1$ .

$f^{- 1} (2 {(x - 6)}^{\frac{1}{3}} + 10) = x + 15 {(x - 6)}^{\frac{2}{3}} + 75 {(x - 6)}^{\frac{1}{3}} + 119 - (15 ({(x - 6)}^{\frac{2}{3}} + {(x - 6)}^{\frac{1}{3}} \cdot 5 + 5 {(x - 6)}^{\frac{1}{3}} + 5 \cdot 5)) + \frac{75 (2 {(x - 6)}^{\frac{1}{3}} + 10)}{2} - 119$

Step 5.2.3.12.1.2

Move $5$ to the left of ${(x - 6)}^{\frac{1}{3}}$ .

$f^{- 1} (2 {(x - 6)}^{\frac{1}{3}} + 10) = x + 15 {(x - 6)}^{\frac{2}{3}} + 75 {(x - 6)}^{\frac{1}{3}} + 119 - (15 ({(x - 6)}^{\frac{2}{3}} + 5 \cdot {(x - 6)}^{\frac{1}{3}} + 5 {(x - 6)}^{\frac{1}{3}} + 5 \cdot 5)) + \frac{75 (2 {(x - 6)}^{\frac{1}{3}} + 10)}{2} - 119$

Step 5.2.3.12.1.3

Multiply $5$ by $5$ .

$f^{- 1} (2 {(x - 6)}^{\frac{1}{3}} + 10) = x + 15 {(x - 6)}^{\frac{2}{3}} + 75 {(x - 6)}^{\frac{1}{3}} + 119 - (15 ({(x - 6)}^{\frac{2}{3}} + 5 {(x - 6)}^{\frac{1}{3}} + 5 {(x - 6)}^{\frac{1}{3}} + 25)) + \frac{75 (2 {(x - 6)}^{\frac{1}{3}} + 10)}{2} - 119$

Step 5.2.3.12.2

Add $5 {(x - 6)}^{\frac{1}{3}}$ and $5 {(x - 6)}^{\frac{1}{3}}$ .

$f^{- 1} (2 {(x - 6)}^{\frac{1}{3}} + 10) = x + 15 {(x - 6)}^{\frac{2}{3}} + 75 {(x - 6)}^{\frac{1}{3}} + 119 - (15 ({(x - 6)}^{\frac{2}{3}} + 10 {(x - 6)}^{\frac{1}{3}} + 25)) + \frac{75 (2 {(x - 6)}^{\frac{1}{3}} + 10)}{2} - 119$

Step 5.2.3.13

Apply the distributive property.

$f^{- 1} (2 {(x - 6)}^{\frac{1}{3}} + 10) = x + 15 {(x - 6)}^{\frac{2}{3}} + 75 {(x - 6)}^{\frac{1}{3}} + 119 - (15 {(x - 6)}^{\frac{2}{3}} + 15 (10 {(x - 6)}^{\frac{1}{3}}) + 15 \cdot 25) + \frac{75 (2 {(x - 6)}^{\frac{1}{3}} + 10)}{2} - 119$

Step 5.2.3.14

Simplify.

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Step 5.2.3.14.1

Multiply $10$ by $15$ .

$f^{- 1} (2 {(x - 6)}^{\frac{1}{3}} + 10) = x + 15 {(x - 6)}^{\frac{2}{3}} + 75 {(x - 6)}^{\frac{1}{3}} + 119 - (15 {(x - 6)}^{\frac{2}{3}} + 150 {(x - 6)}^{\frac{1}{3}} + 15 \cdot 25) + \frac{75 (2 {(x - 6)}^{\frac{1}{3}} + 10)}{2} - 119$

Step 5.2.3.14.2

Multiply $15$ by $25$ .

$f^{- 1} (2 {(x - 6)}^{\frac{1}{3}} + 10) = x + 15 {(x - 6)}^{\frac{2}{3}} + 75 {(x - 6)}^{\frac{1}{3}} + 119 - (15 {(x - 6)}^{\frac{2}{3}} + 150 {(x - 6)}^{\frac{1}{3}} + 375) + \frac{75 (2 {(x - 6)}^{\frac{1}{3}} + 10)}{2} - 119$

Step 5.2.3.15

Apply the distributive property.

$f^{- 1} (2 {(x - 6)}^{\frac{1}{3}} + 10) = x + 15 {(x - 6)}^{\frac{2}{3}} + 75 {(x - 6)}^{\frac{1}{3}} + 119 - (15 {(x - 6)}^{\frac{2}{3}}) - (150 {(x - 6)}^{\frac{1}{3}}) - 1 \cdot 375 + \frac{75 (2 {(x - 6)}^{\frac{1}{3}} + 10)}{2} - 119$

Step 5.2.3.16

Simplify.

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Step 5.2.3.16.1

Multiply $15$ by $- 1$ .

$f^{- 1} (2 {(x - 6)}^{\frac{1}{3}} + 10) = x + 15 {(x - 6)}^{\frac{2}{3}} + 75 {(x - 6)}^{\frac{1}{3}} + 119 - 15 {(x - 6)}^{\frac{2}{3}} - (150 {(x - 6)}^{\frac{1}{3}}) - 1 \cdot 375 + \frac{75 (2 {(x - 6)}^{\frac{1}{3}} + 10)}{2} - 119$

Step 5.2.3.16.2

Multiply $150$ by $- 1$ .

$f^{- 1} (2 {(x - 6)}^{\frac{1}{3}} + 10) = x + 15 {(x - 6)}^{\frac{2}{3}} + 75 {(x - 6)}^{\frac{1}{3}} + 119 - 15 {(x - 6)}^{\frac{2}{3}} - 150 {(x - 6)}^{\frac{1}{3}} - 1 \cdot 375 + \frac{75 (2 {(x - 6)}^{\frac{1}{3}} + 10)}{2} - 119$

Step 5.2.3.16.3

Multiply $- 1$ by $375$ .

$f^{- 1} (2 {(x - 6)}^{\frac{1}{3}} + 10) = x + 15 {(x - 6)}^{\frac{2}{3}} + 75 {(x - 6)}^{\frac{1}{3}} + 119 - 15 {(x - 6)}^{\frac{2}{3}} - 150 {(x - 6)}^{\frac{1}{3}} - 375 + \frac{75 (2 {(x - 6)}^{\frac{1}{3}} + 10)}{2} - 119$

Step 5.2.3.17

Factor $2$ out of $75 (2 {(x - 6)}^{\frac{1}{3}} + 10)$ .

$f^{- 1} (2 {(x - 6)}^{\frac{1}{3}} + 10) = x + 15 {(x - 6)}^{\frac{2}{3}} + 75 {(x - 6)}^{\frac{1}{3}} + 119 - 15 {(x - 6)}^{\frac{2}{3}} - 150 {(x - 6)}^{\frac{1}{3}} - 375 + \frac{2 (75 ({(x - 6)}^{\frac{1}{3}} + 5))}{2} - 119$

Step 5.2.3.18

Cancel the common factors.

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Step 5.2.3.18.1

Factor $2$ out of $2$ .

Step 5.2.3.18.2

Cancel the common factor.

Step 5.2.3.18.3

Rewrite the expression.

Step 5.2.3.18.4

Divide $75 ({(x - 6)}^{\frac{1}{3}} + 5)$ by $1$ .

$f^{- 1} (2 {(x - 6)}^{\frac{1}{3}} + 10) = x + 15 {(x - 6)}^{\frac{2}{3}} + 75 {(x - 6)}^{\frac{1}{3}} + 119 - 15 {(x - 6)}^{\frac{2}{3}} - 150 {(x - 6)}^{\frac{1}{3}} - 375 + 75 ({(x - 6)}^{\frac{1}{3}} + 5) - 119$

Step 5.2.3.19

Apply the distributive property.

$f^{- 1} (2 {(x - 6)}^{\frac{1}{3}} + 10) = x + 15 {(x - 6)}^{\frac{2}{3}} + 75 {(x - 6)}^{\frac{1}{3}} + 119 - 15 {(x - 6)}^{\frac{2}{3}} - 150 {(x - 6)}^{\frac{1}{3}} - 375 + 75 {(x - 6)}^{\frac{1}{3}} + 75 \cdot 5 - 119$

Step 5.2.3.20

Multiply $75$ by $5$ .

Step 5.2.4

Simplify by adding terms.

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Step 5.2.4.1

Combine the opposite terms in $x + 15 {(x - 6)}^{\frac{2}{3}} + 75 {(x - 6)}^{\frac{1}{3}} + 119 - 15 {(x - 6)}^{\frac{2}{3}} - 150 {(x - 6)}^{\frac{1}{3}} - 375 + 75 {(x - 6)}^{\frac{1}{3}} + 375 - 119$ .

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Step 5.2.4.1.1

Subtract $15 {(x - 6)}^{\frac{2}{3}}$ from $15 {(x - 6)}^{\frac{2}{3}}$ .

$f^{- 1} (2 {(x - 6)}^{\frac{1}{3}} + 10) = x + 75 {(x - 6)}^{\frac{1}{3}} + 119 + 0 - 150 {(x - 6)}^{\frac{1}{3}} - 375 + 75 {(x - 6)}^{\frac{1}{3}} + 375 - 119$

Step 5.2.4.1.2

Add $x + 75 {(x - 6)}^{\frac{1}{3}} + 119$ and $0$ .

$f^{- 1} (2 {(x - 6)}^{\frac{1}{3}} + 10) = x + 75 {(x - 6)}^{\frac{1}{3}} + 119 - 150 {(x - 6)}^{\frac{1}{3}} - 375 + 75 {(x - 6)}^{\frac{1}{3}} + 375 - 119$

Step 5.2.4.1.3

Add $- 375$ and $375$ .

$f^{- 1} (2 {(x - 6)}^{\frac{1}{3}} + 10) = x + 75 {(x - 6)}^{\frac{1}{3}} + 119 - 150 {(x - 6)}^{\frac{1}{3}} + 75 {(x - 6)}^{\frac{1}{3}} + 0 - 119$

Step 5.2.4.1.4

Add $x + 75 {(x - 6)}^{\frac{1}{3}} + 119 - 150 {(x - 6)}^{\frac{1}{3}} + 75 {(x - 6)}^{\frac{1}{3}}$ and $0$ .

$f^{- 1} (2 {(x - 6)}^{\frac{1}{3}} + 10) = x + 75 {(x - 6)}^{\frac{1}{3}} + 119 - 150 {(x - 6)}^{\frac{1}{3}} + 75 {(x - 6)}^{\frac{1}{3}} - 119$

Step 5.2.4.1.5

Subtract $119$ from $119$ .

$f^{- 1} (2 {(x - 6)}^{\frac{1}{3}} + 10) = x + 75 {(x - 6)}^{\frac{1}{3}} + 0 - 150 {(x - 6)}^{\frac{1}{3}} + 75 {(x - 6)}^{\frac{1}{3}}$

Step 5.2.4.1.6

Add $x + 75 {(x - 6)}^{\frac{1}{3}}$ and $0$ .

$f^{- 1} (2 {(x - 6)}^{\frac{1}{3}} + 10) = x + 75 {(x - 6)}^{\frac{1}{3}} - 150 {(x - 6)}^{\frac{1}{3}} + 75 {(x - 6)}^{\frac{1}{3}}$

Step 5.2.4.2

Subtract $150 {(x - 6)}^{\frac{1}{3}}$ from $75 {(x - 6)}^{\frac{1}{3}}$ .

$f^{- 1} (2 {(x - 6)}^{\frac{1}{3}} + 10) = x - 75 {(x - 6)}^{\frac{1}{3}} + 75 {(x - 6)}^{\frac{1}{3}}$

Step 5.2.4.3

Combine the opposite terms in $x - 75 {(x - 6)}^{\frac{1}{3}} + 75 {(x - 6)}^{\frac{1}{3}}$ .

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Step 5.2.4.3.1

Add $- 75 {(x - 6)}^{\frac{1}{3}}$ and $75 {(x - 6)}^{\frac{1}{3}}$ .

$f^{- 1} (2 {(x - 6)}^{\frac{1}{3}} + 10) = x + 0$

Step 5.2.4.3.2

Add $x$ and $0$ .

$f^{- 1} (2 {(x - 6)}^{\frac{1}{3}} + 10) = x$

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 $f (\frac{x^{3}}{8} - \frac{15 x^{2}}{4} + \frac{75 x}{2} - 119)$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\frac{x^{3}}{8} - \frac{15 x^{2}}{4} + \frac{75 x}{2} - 119) = 2 {((\frac{x^{3}}{8} - \frac{15 x^{2}}{4} + \frac{75 x}{2} - 119) - 6)}^{\frac{1}{3}} + 10$

Step 5.3.3

Subtract $6$ from $- 119$ .

$f (\frac{x^{3}}{8} - \frac{15 x^{2}}{4} + \frac{75 x}{2} - 119) = 2 {(\frac{x^{3}}{8} - \frac{15 x^{2}}{4} + \frac{75 x}{2} - 125)}^{\frac{1}{3}} + 10$

Step 5.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{x^{3}}{8} - \frac{15 x^{2}}{4} + \frac{75 x}{2} - 119$ is the inverse of $f (x) = 2 {(x - 6)}^{\frac{1}{3}} + 10$ .

$f^{- 1} (x) = \frac{x^{3}}{8} - \frac{15 x^{2}}{4} + \frac{75 x}{2} - 119$