Precalculus Examples

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

Step 1

Write $A (x) = x^{\frac{1}{3}} - 3$ as an equation.

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

Step 2

Interchange the variables.

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

Step 3

Solve for $y$ .

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Step 3.1

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

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

Step 3.2

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

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Step 3.2.1

Add $3$ to both sides of the equation.

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

Step 3.2.2

Add $x$ to both sides of the equation.

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

Step 3.3

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

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

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

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Step 3.4.1.1.1

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

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Step 3.4.1.1.1.1

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

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

Step 3.4.1.1.1.2

Cancel the common factor of $3$ .

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Step 3.4.1.1.1.2.1

Cancel the common factor.

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

Step 3.4.1.1.1.2.2

Rewrite the expression.

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

Step 3.4.1.1.2

Simplify.

$y = {(3 + x)}^{3}$

Step 3.4.2

Simplify the right side.

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Step 3.4.2.1

Simplify ${(3 + x)}^{3}$ .

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Step 3.4.2.1.1

Use the Binomial Theorem.

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

Step 3.4.2.1.2

Simplify each term.

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Step 3.4.2.1.2.1

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

$y = 27 + 3 \cdot 3^{2} x + 3 \cdot 3 x^{2} + x^{3}$

Step 3.4.2.1.2.2

Multiply $3$ by $3^{2}$ by adding the exponents.

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Step 3.4.2.1.2.2.1

Multiply $3$ by $3^{2}$ .

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Step 3.4.2.1.2.2.1.1

Raise $3$ to the power of $1$ .

$y = 27 + 3^{1} \cdot 3^{2} x + 3 \cdot 3 x^{2} + x^{3}$

Step 3.4.2.1.2.2.1.2

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

$y = 27 + 3^{1 + 2} x + 3 \cdot 3 x^{2} + x^{3}$

Step 3.4.2.1.2.2.2

Add $1$ and $2$ .

$y = 27 + 3^{3} x + 3 \cdot 3 x^{2} + x^{3}$

Step 3.4.2.1.2.3

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

$y = 27 + 27 x + 3 \cdot 3 x^{2} + x^{3}$

Step 3.4.2.1.2.4

Multiply $3$ by $3$ .

$y = 27 + 27 x + 9 x^{2} + x^{3}$

Step 3.5

Simplify $27 + 27 x + 9 x^{2} + x^{3}$ .

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Step 3.5.1

Move $27$ .

$y = 27 x + 9 x^{2} + x^{3} + 27$

Step 3.5.2

Move $27 x$ .

$y = 9 x^{2} + x^{3} + 27 x + 27$

Step 3.5.3

Reorder $9 x^{2}$ and $x^{3}$ .

$y = x^{3} + 9 x^{2} + 27 x + 27$

Step 4

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

$A^{- 1} (x) = x^{3} + 9 x^{2} + 27 x + 27$

Step 5

Verify if $A^{- 1} (x) = x^{3} + 9 x^{2} + 27 x + 27$ is the inverse of $A (x) = x^{\frac{1}{3}} - 3$ .

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Step 5.1

To verify the inverse, check if $A^{- 1} (A (x)) = x$ and $A (A^{- 1} (x)) = x$ .

Step 5.2

Evaluate $A^{- 1} (A (x))$ .

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Step 5.2.1

Set up the composite result function.

$A^{- 1} (A (x))$

Step 5.2.2

Evaluate $A^{- 1} (x^{\frac{1}{3}} - 3)$ by substituting in the value of $A$ into $A^{- 1}$ .

$A^{- 1} (x^{\frac{1}{3}} - 3) = {(x^{\frac{1}{3}} - 3)}^{3} + 9 {(x^{\frac{1}{3}} - 3)}^{2} + 27 (x^{\frac{1}{3}} - 3) + 27$

Step 5.2.3

Simplify each term.

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Step 5.2.3.1

Use the Binomial Theorem.

$A^{- 1} (x^{\frac{1}{3}} - 3) = {(x^{\frac{1}{3}})}^{3} + 3 {(x^{\frac{1}{3}})}^{2} \cdot - 3 + 3 x^{\frac{1}{3}} {(- 3)}^{2} + {(- 3)}^{3} + 9 {(x^{\frac{1}{3}} - 3)}^{2} + 27 (x^{\frac{1}{3}} - 3) + 27$

Step 5.2.3.2

Simplify each term.

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Step 5.2.3.2.1

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

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Step 5.2.3.2.1.1

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

$A^{- 1} (x^{\frac{1}{3}} - 3) = x^{\frac{1}{3} \cdot 3} + 3 {(x^{\frac{1}{3}})}^{2} \cdot - 3 + 3 x^{\frac{1}{3}} {(- 3)}^{2} + {(- 3)}^{3} + 9 {(x^{\frac{1}{3}} - 3)}^{2} + 27 (x^{\frac{1}{3}} - 3) + 27$

Step 5.2.3.2.1.2

Cancel the common factor of $3$ .

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Step 5.2.3.2.1.2.1

Cancel the common factor.

Step 5.2.3.2.1.2.2

Rewrite the expression.

$A^{- 1} (x^{\frac{1}{3}} - 3) = x + 3 {(x^{\frac{1}{3}})}^{2} \cdot - 3 + 3 x^{\frac{1}{3}} {(- 3)}^{2} + {(- 3)}^{3} + 9 {(x^{\frac{1}{3}} - 3)}^{2} + 27 (x^{\frac{1}{3}} - 3) + 27$

Step 5.2.3.2.2

Simplify.

$A^{- 1} (x^{\frac{1}{3}} - 3) = x + 3 {(x^{\frac{1}{3}})}^{2} \cdot - 3 + 3 x^{\frac{1}{3}} {(- 3)}^{2} + {(- 3)}^{3} + 9 {(x^{\frac{1}{3}} - 3)}^{2} + 27 (x^{\frac{1}{3}} - 3) + 27$

Step 5.2.3.2.3

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

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Step 5.2.3.2.3.1

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

$A^{- 1} (x^{\frac{1}{3}} - 3) = x + 3 x^{\frac{1}{3} \cdot 2} \cdot - 3 + 3 x^{\frac{1}{3}} {(- 3)}^{2} + {(- 3)}^{3} + 9 {(x^{\frac{1}{3}} - 3)}^{2} + 27 (x^{\frac{1}{3}} - 3) + 27$

Step 5.2.3.2.3.2

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

$A^{- 1} (x^{\frac{1}{3}} - 3) = x + 3 x^{\frac{2}{3}} \cdot - 3 + 3 x^{\frac{1}{3}} {(- 3)}^{2} + {(- 3)}^{3} + 9 {(x^{\frac{1}{3}} - 3)}^{2} + 27 (x^{\frac{1}{3}} - 3) + 27$

Step 5.2.3.2.4

Multiply $- 3$ by $3$ .

$A^{- 1} (x^{\frac{1}{3}} - 3) = x - 9 x^{\frac{2}{3}} + 3 x^{\frac{1}{3}} {(- 3)}^{2} + {(- 3)}^{3} + 9 {(x^{\frac{1}{3}} - 3)}^{2} + 27 (x^{\frac{1}{3}} - 3) + 27$

Step 5.2.3.2.5

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

$A^{- 1} (x^{\frac{1}{3}} - 3) = x - 9 x^{\frac{2}{3}} + 3 x^{\frac{1}{3}} \cdot 9 + {(- 3)}^{3} + 9 {(x^{\frac{1}{3}} - 3)}^{2} + 27 (x^{\frac{1}{3}} - 3) + 27$

Step 5.2.3.2.6

Multiply $9$ by $3$ .

$A^{- 1} (x^{\frac{1}{3}} - 3) = x - 9 x^{\frac{2}{3}} + 27 x^{\frac{1}{3}} + {(- 3)}^{3} + 9 {(x^{\frac{1}{3}} - 3)}^{2} + 27 (x^{\frac{1}{3}} - 3) + 27$

Step 5.2.3.2.7

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

$A^{- 1} (x^{\frac{1}{3}} - 3) = x - 9 x^{\frac{2}{3}} + 27 x^{\frac{1}{3}} - 27 + 9 {(x^{\frac{1}{3}} - 3)}^{2} + 27 (x^{\frac{1}{3}} - 3) + 27$

Step 5.2.3.3

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

$A^{- 1} (x^{\frac{1}{3}} - 3) = x - 9 x^{\frac{2}{3}} + 27 x^{\frac{1}{3}} - 27 + 9 ((x^{\frac{1}{3}} - 3) (x^{\frac{1}{3}} - 3)) + 27 (x^{\frac{1}{3}} - 3) + 27$

Step 5.2.3.4

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

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Step 5.2.3.4.1

Apply the distributive property.

$A^{- 1} (x^{\frac{1}{3}} - 3) = x - 9 x^{\frac{2}{3}} + 27 x^{\frac{1}{3}} - 27 + 9 (x^{\frac{1}{3}} (x^{\frac{1}{3}} - 3) - 3 (x^{\frac{1}{3}} - 3)) + 27 (x^{\frac{1}{3}} - 3) + 27$

Step 5.2.3.4.2

Apply the distributive property.

$A^{- 1} (x^{\frac{1}{3}} - 3) = x - 9 x^{\frac{2}{3}} + 27 x^{\frac{1}{3}} - 27 + 9 (x^{\frac{1}{3}} x^{\frac{1}{3}} + x^{\frac{1}{3}} \cdot - 3 - 3 (x^{\frac{1}{3}} - 3)) + 27 (x^{\frac{1}{3}} - 3) + 27$

Step 5.2.3.4.3

Apply the distributive property.

$A^{- 1} (x^{\frac{1}{3}} - 3) = x - 9 x^{\frac{2}{3}} + 27 x^{\frac{1}{3}} - 27 + 9 (x^{\frac{1}{3}} x^{\frac{1}{3}} + x^{\frac{1}{3}} \cdot - 3 - 3 x^{\frac{1}{3}} - 3 \cdot - 3) + 27 (x^{\frac{1}{3}} - 3) + 27$

Step 5.2.3.5

Simplify and combine like terms.

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Step 5.2.3.5.1

Simplify each term.

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Step 5.2.3.5.1.1

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

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Step 5.2.3.5.1.1.1

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

$A^{- 1} (x^{\frac{1}{3}} - 3) = x - 9 x^{\frac{2}{3}} + 27 x^{\frac{1}{3}} - 27 + 9 (x^{\frac{1}{3} + \frac{1}{3}} + x^{\frac{1}{3}} \cdot - 3 - 3 x^{\frac{1}{3}} - 3 \cdot - 3) + 27 (x^{\frac{1}{3}} - 3) + 27$

Step 5.2.3.5.1.1.2

Combine the numerators over the common denominator.

$A^{- 1} (x^{\frac{1}{3}} - 3) = x - 9 x^{\frac{2}{3}} + 27 x^{\frac{1}{3}} - 27 + 9 (x^{\frac{1 + 1}{3}} + x^{\frac{1}{3}} \cdot - 3 - 3 x^{\frac{1}{3}} - 3 \cdot - 3) + 27 (x^{\frac{1}{3}} - 3) + 27$

Step 5.2.3.5.1.1.3

Add $1$ and $1$ .

$A^{- 1} (x^{\frac{1}{3}} - 3) = x - 9 x^{\frac{2}{3}} + 27 x^{\frac{1}{3}} - 27 + 9 (x^{\frac{2}{3}} + x^{\frac{1}{3}} \cdot - 3 - 3 x^{\frac{1}{3}} - 3 \cdot - 3) + 27 (x^{\frac{1}{3}} - 3) + 27$

Step 5.2.3.5.1.2

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

$A^{- 1} (x^{\frac{1}{3}} - 3) = x - 9 x^{\frac{2}{3}} + 27 x^{\frac{1}{3}} - 27 + 9 (x^{\frac{2}{3}} - 3 \cdot x^{\frac{1}{3}} - 3 x^{\frac{1}{3}} - 3 \cdot - 3) + 27 (x^{\frac{1}{3}} - 3) + 27$

Step 5.2.3.5.1.3

Multiply $- 3$ by $- 3$ .

$A^{- 1} (x^{\frac{1}{3}} - 3) = x - 9 x^{\frac{2}{3}} + 27 x^{\frac{1}{3}} - 27 + 9 (x^{\frac{2}{3}} - 3 x^{\frac{1}{3}} - 3 x^{\frac{1}{3}} + 9) + 27 (x^{\frac{1}{3}} - 3) + 27$

Step 5.2.3.5.2

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

$A^{- 1} (x^{\frac{1}{3}} - 3) = x - 9 x^{\frac{2}{3}} + 27 x^{\frac{1}{3}} - 27 + 9 (x^{\frac{2}{3}} - 6 x^{\frac{1}{3}} + 9) + 27 (x^{\frac{1}{3}} - 3) + 27$

Step 5.2.3.6

Apply the distributive property.

$A^{- 1} (x^{\frac{1}{3}} - 3) = x - 9 x^{\frac{2}{3}} + 27 x^{\frac{1}{3}} - 27 + 9 x^{\frac{2}{3}} + 9 (- 6 x^{\frac{1}{3}}) + 9 \cdot 9 + 27 (x^{\frac{1}{3}} - 3) + 27$

Step 5.2.3.7

Simplify.

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Step 5.2.3.7.1

Multiply $- 6$ by $9$ .

$A^{- 1} (x^{\frac{1}{3}} - 3) = x - 9 x^{\frac{2}{3}} + 27 x^{\frac{1}{3}} - 27 + 9 x^{\frac{2}{3}} - 54 x^{\frac{1}{3}} + 9 \cdot 9 + 27 (x^{\frac{1}{3}} - 3) + 27$

Step 5.2.3.7.2

Multiply $9$ by $9$ .

$A^{- 1} (x^{\frac{1}{3}} - 3) = x - 9 x^{\frac{2}{3}} + 27 x^{\frac{1}{3}} - 27 + 9 x^{\frac{2}{3}} - 54 x^{\frac{1}{3}} + 81 + 27 (x^{\frac{1}{3}} - 3) + 27$

Step 5.2.3.8

Apply the distributive property.

$A^{- 1} (x^{\frac{1}{3}} - 3) = x - 9 x^{\frac{2}{3}} + 27 x^{\frac{1}{3}} - 27 + 9 x^{\frac{2}{3}} - 54 x^{\frac{1}{3}} + 81 + 27 x^{\frac{1}{3}} + 27 \cdot - 3 + 27$

Step 5.2.3.9

Multiply $27$ by $- 3$ .

$A^{- 1} (x^{\frac{1}{3}} - 3) = x - 9 x^{\frac{2}{3}} + 27 x^{\frac{1}{3}} - 27 + 9 x^{\frac{2}{3}} - 54 x^{\frac{1}{3}} + 81 + 27 x^{\frac{1}{3}} - 81 + 27$

Step 5.2.4

Simplify by adding terms.

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Step 5.2.4.1

Combine the opposite terms in $x - 9 x^{\frac{2}{3}} + 27 x^{\frac{1}{3}} - 27 + 9 x^{\frac{2}{3}} - 54 x^{\frac{1}{3}} + 81 + 27 x^{\frac{1}{3}} - 81 + 27$ .

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Step 5.2.4.1.1

Add $- 9 x^{\frac{2}{3}}$ and $9 x^{\frac{2}{3}}$ .

$A^{- 1} (x^{\frac{1}{3}} - 3) = x + 27 x^{\frac{1}{3}} - 27 + 0 - 54 x^{\frac{1}{3}} + 81 + 27 x^{\frac{1}{3}} - 81 + 27$

Step 5.2.4.1.2

Add $x + 27 x^{\frac{1}{3}} - 27$ and $0$ .

$A^{- 1} (x^{\frac{1}{3}} - 3) = x + 27 x^{\frac{1}{3}} - 27 - 54 x^{\frac{1}{3}} + 81 + 27 x^{\frac{1}{3}} - 81 + 27$

Step 5.2.4.1.3

Subtract $81$ from $81$ .

$A^{- 1} (x^{\frac{1}{3}} - 3) = x + 27 x^{\frac{1}{3}} - 27 - 54 x^{\frac{1}{3}} + 27 x^{\frac{1}{3}} + 0 + 27$

Step 5.2.4.1.4

Add $x + 27 x^{\frac{1}{3}} - 27 - 54 x^{\frac{1}{3}} + 27 x^{\frac{1}{3}}$ and $0$ .

$A^{- 1} (x^{\frac{1}{3}} - 3) = x + 27 x^{\frac{1}{3}} - 27 - 54 x^{\frac{1}{3}} + 27 x^{\frac{1}{3}} + 27$

Step 5.2.4.1.5

Add $- 27$ and $27$ .

$A^{- 1} (x^{\frac{1}{3}} - 3) = x + 27 x^{\frac{1}{3}} - 54 x^{\frac{1}{3}} + 27 x^{\frac{1}{3}} + 0$

Step 5.2.4.1.6

Add $x + 27 x^{\frac{1}{3}} - 54 x^{\frac{1}{3}} + 27 x^{\frac{1}{3}}$ and $0$ .

$A^{- 1} (x^{\frac{1}{3}} - 3) = x + 27 x^{\frac{1}{3}} - 54 x^{\frac{1}{3}} + 27 x^{\frac{1}{3}}$

Step 5.2.4.2

Subtract $54 x^{\frac{1}{3}}$ from $27 x^{\frac{1}{3}}$ .

$A^{- 1} (x^{\frac{1}{3}} - 3) = x - 27 x^{\frac{1}{3}} + 27 x^{\frac{1}{3}}$

Step 5.2.4.3

Combine the opposite terms in $x - 27 x^{\frac{1}{3}} + 27 x^{\frac{1}{3}}$ .

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Step 5.2.4.3.1

Add $- 27 x^{\frac{1}{3}}$ and $27 x^{\frac{1}{3}}$ .

$A^{- 1} (x^{\frac{1}{3}} - 3) = x + 0$

Step 5.2.4.3.2

Add $x$ and $0$ .

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

Step 5.3

Evaluate $A (A^{- 1} (x))$ .

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Step 5.3.1

Set up the composite result function.

$A (A^{- 1} (x))$

Step 5.3.2

Evaluate $A (x^{3} + 9 x^{2} + 27 x + 27)$ by substituting in the value of $A^{- 1}$ into $A$ .

$A (x^{3} + 9 x^{2} + 27 x + 27) = {(x^{3} + 9 x^{2} + 27 x + 27)}^{\frac{1}{3}} - 3$

Step 5.4

Since $A^{- 1} (A (x)) = x$ and $A (A^{- 1} (x)) = x$ , then $A^{- 1} (x) = x^{3} + 9 x^{2} + 27 x + 27$ is the inverse of $A (x) = x^{\frac{1}{3}} - 3$ .

$A^{- 1} (x) = x^{3} + 9 x^{2} + 27 x + 27$