Precalculus Examples

$v (x) = \frac{\sqrt{- x}}{3} - 3$

Step 1

Write $v (x) = \frac{\sqrt{- x}}{3} - 3$ as an equation.

$y = \frac{\sqrt{- x}}{3} - 3$

Step 2

Interchange the variables.

$x = \frac{\sqrt{- y}}{3} - 3$

Step 3

Solve for $y$ .

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Step 3.1

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

$\frac{\sqrt{- y}}{3} - 3 = x$

Step 3.2

Solve for $\sqrt{- y}$ .

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Step 3.2.1

Add $3$ to both sides of the equation.

$\frac{\sqrt{- y}}{3} = x + 3$

Step 3.2.2

Multiply both sides of the equation by $3$ .

$3 \frac{\sqrt{- y}}{3} = 3 (x + 3)$

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 $3$ .

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Step 3.2.3.1.1.1

Cancel the common factor.

$3 \frac{\sqrt{- y}}{3} = 3 (x + 3)$

Step 3.2.3.1.1.2

Rewrite the expression.

$\sqrt{- y} = 3 (x + 3)$

Step 3.2.3.2

Simplify the right side.

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Step 3.2.3.2.1

Simplify $3 (x + 3)$ .

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Step 3.2.3.2.1.1

Apply the distributive property.

$\sqrt{- y} = 3 x + 3 \cdot 3$

Step 3.2.3.2.1.2

Multiply $3$ by $3$ .

$\sqrt{- y} = 3 x + 9$

Step 3.3

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

${\sqrt{- y}}^{2} = {(3 x + 9)}^{2}$

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 $\sqrt{- y}$ as ${(- y)}^{\frac{1}{2}}$ .

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

Step 3.4.2

Simplify the left side.

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Step 3.4.2.1

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

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Step 3.4.2.1.1

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

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Step 3.4.2.1.1.1

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

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

Step 3.4.2.1.1.2

Cancel the common factor of $2$ .

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Step 3.4.2.1.1.2.1

Cancel the common factor.

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

Step 3.4.2.1.1.2.2

Rewrite the expression.

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

Step 3.4.2.1.2

Simplify.

$- y = {(3 x + 9)}^{2}$

Step 3.4.3

Simplify the right side.

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Step 3.4.3.1

Simplify ${(3 x + 9)}^{2}$ .

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Step 3.4.3.1.1

Rewrite ${(3 x + 9)}^{2}$ as $(3 x + 9) (3 x + 9)$ .

$- y = (3 x + 9) (3 x + 9)$

Step 3.4.3.1.2

Expand $(3 x + 9) (3 x + 9)$ using the FOIL Method.

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Step 3.4.3.1.2.1

Apply the distributive property.

$- y = 3 x (3 x + 9) + 9 (3 x + 9)$

Step 3.4.3.1.2.2

Apply the distributive property.

$- y = 3 x (3 x) + 3 x \cdot 9 + 9 (3 x + 9)$

Step 3.4.3.1.2.3

Apply the distributive property.

$- y = 3 x (3 x) + 3 x \cdot 9 + 9 (3 x) + 9 \cdot 9$

Step 3.4.3.1.3

Simplify and combine like terms.

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Step 3.4.3.1.3.1

Simplify each term.

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Step 3.4.3.1.3.1.1

Rewrite using the commutative property of multiplication.

$- y = 3 \cdot 3 x \cdot x + 3 x \cdot 9 + 9 (3 x) + 9 \cdot 9$

Step 3.4.3.1.3.1.2

Multiply $x$ by $x$ by adding the exponents.

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Step 3.4.3.1.3.1.2.1

Move $x$ .

$- y = 3 \cdot 3 (x \cdot x) + 3 x \cdot 9 + 9 (3 x) + 9 \cdot 9$

Step 3.4.3.1.3.1.2.2

Multiply $x$ by $x$ .

$- y = 3 \cdot 3 x^{2} + 3 x \cdot 9 + 9 (3 x) + 9 \cdot 9$

Step 3.4.3.1.3.1.3

Multiply $3$ by $3$ .

$- y = 9 x^{2} + 3 x \cdot 9 + 9 (3 x) + 9 \cdot 9$

Step 3.4.3.1.3.1.4

Multiply $9$ by $3$ .

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

Step 3.4.3.1.3.1.5

Multiply $3$ by $9$ .

$- y = 9 x^{2} + 27 x + 27 x + 9 \cdot 9$

Step 3.4.3.1.3.1.6

Multiply $9$ by $9$ .

$- y = 9 x^{2} + 27 x + 27 x + 81$

Step 3.4.3.1.3.2

Add $27 x$ and $27 x$ .

$- y = 9 x^{2} + 54 x + 81$

Step 3.5

Divide each term in $- y = 9 x^{2} + 54 x + 81$ by $- 1$ and simplify.

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Step 3.5.1

Divide each term in $- y = 9 x^{2} + 54 x + 81$ by $- 1$ .

$\frac{- y}{- 1} = \frac{9 x^{2}}{- 1} + \frac{54 x}{- 1} + \frac{81}{- 1}$

Step 3.5.2

Simplify the left side.

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Step 3.5.2.1

Dividing two negative values results in a positive value.

$\frac{y}{1} = \frac{9 x^{2}}{- 1} + \frac{54 x}{- 1} + \frac{81}{- 1}$

Step 3.5.2.2

Divide $y$ by $1$ .

$y = \frac{9 x^{2}}{- 1} + \frac{54 x}{- 1} + \frac{81}{- 1}$

Step 3.5.3

Simplify the right side.

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Step 3.5.3.1

Simplify each term.

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Step 3.5.3.1.1

Move the negative one from the denominator of $\frac{9 x^{2}}{- 1}$ .

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

Step 3.5.3.1.2

Rewrite $- 1 \cdot (9 x^{2})$ as $- (9 x^{2})$ .

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

Step 3.5.3.1.3

Multiply $9$ by $- 1$ .

$y = - 9 x^{2} + \frac{54 x}{- 1} + \frac{81}{- 1}$

Step 3.5.3.1.4

Move the negative one from the denominator of $\frac{54 x}{- 1}$ .

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

Step 3.5.3.1.5

Rewrite $- 1 \cdot (54 x)$ as $- (54 x)$ .

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

Step 3.5.3.1.6

Multiply $54$ by $- 1$ .

$y = - 9 x^{2} - 54 x + \frac{81}{- 1}$

Step 3.5.3.1.7

Divide $81$ by $- 1$ .

$y = - 9 x^{2} - 54 x - 81$

Step 4

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

$v^{- 1} (x) = - 9 x^{2} - 54 x - 81$

Step 5

Verify if $v^{- 1} (x) = - 9 x^{2} - 54 x - 81$ is the inverse of $v (x) = \frac{\sqrt{- x}}{3} - 3$ .

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Step 5.1

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

Step 5.2

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

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Step 5.2.1

Set up the composite result function.

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

Step 5.2.2

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

$v^{- 1} (\frac{\sqrt{- x}}{3} - 3) = - 9 {(\frac{\sqrt{- x}}{3} - 3)}^{2} - 54 (\frac{\sqrt{- x}}{3} - 3) - 81$

Step 5.2.3

Simplify each term.

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Step 5.2.3.1

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

$v^{- 1} (\frac{\sqrt{- x}}{3} - 3) = - 9 ((\frac{\sqrt{- x}}{3} - 3) (\frac{\sqrt{- x}}{3} - 3)) - 54 (\frac{\sqrt{- x}}{3} - 3) - 81$

Step 5.2.3.2

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

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Step 5.2.3.2.1

Apply the distributive property.

$v^{- 1} (\frac{\sqrt{- x}}{3} - 3) = - 9 (\frac{\sqrt{- x}}{3} \cdot (\frac{\sqrt{- x}}{3} - 3) - 3 (\frac{\sqrt{- x}}{3} - 3)) - 54 (\frac{\sqrt{- x}}{3} - 3) - 81$

Step 5.2.3.2.2

Apply the distributive property.

$v^{- 1} (\frac{\sqrt{- x}}{3} - 3) = - 9 (\frac{\sqrt{- x}}{3} \cdot \frac{\sqrt{- x}}{3} + \frac{\sqrt{- x}}{3} \cdot - 3 - 3 (\frac{\sqrt{- x}}{3} - 3)) - 54 (\frac{\sqrt{- x}}{3} - 3) - 81$

Step 5.2.3.2.3

Apply the distributive property.

$v^{- 1} (\frac{\sqrt{- x}}{3} - 3) = - 9 (\frac{\sqrt{- x}}{3} \cdot \frac{\sqrt{- x}}{3} + \frac{\sqrt{- x}}{3} \cdot - 3 - 3 \frac{\sqrt{- x}}{3} - 3 \cdot - 3) - 54 (\frac{\sqrt{- x}}{3} - 3) - 81$

Step 5.2.3.3

Simplify and combine like terms.

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Step 5.2.3.3.1

Simplify each term.

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Step 5.2.3.3.1.1

Multiply $\frac{\sqrt{- x}}{3} \cdot \frac{\sqrt{- x}}{3}$ .

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Step 5.2.3.3.1.1.1

Multiply $\frac{\sqrt{- x}}{3}$ by $\frac{\sqrt{- x}}{3}$ .

$v^{- 1} (\frac{\sqrt{- x}}{3} - 3) = - 9 (\frac{\sqrt{- x} \sqrt{- x}}{3 \cdot 3} + \frac{\sqrt{- x}}{3} \cdot - 3 - 3 \frac{\sqrt{- x}}{3} - 3 \cdot - 3) - 54 (\frac{\sqrt{- x}}{3} - 3) - 81$

Step 5.2.3.3.1.1.2

Raise $\sqrt{- x}$ to the power of $1$ .

$v^{- 1} (\frac{\sqrt{- x}}{3} - 3) = - 9 (\frac{\sqrt{- x} \sqrt{- x}}{3 \cdot 3} + \frac{\sqrt{- x}}{3} \cdot - 3 - 3 \frac{\sqrt{- x}}{3} - 3 \cdot - 3) - 54 (\frac{\sqrt{- x}}{3} - 3) - 81$

Step 5.2.3.3.1.1.3

Raise $\sqrt{- x}$ to the power of $1$ .

$v^{- 1} (\frac{\sqrt{- x}}{3} - 3) = - 9 (\frac{\sqrt{- x} \sqrt{- x}}{3 \cdot 3} + \frac{\sqrt{- x}}{3} \cdot - 3 - 3 \frac{\sqrt{- x}}{3} - 3 \cdot - 3) - 54 (\frac{\sqrt{- x}}{3} - 3) - 81$

Step 5.2.3.3.1.1.4

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

$v^{- 1} (\frac{\sqrt{- x}}{3} - 3) = - 9 (\frac{{\sqrt{- x}}^{1 + 1}}{3 \cdot 3} + \frac{\sqrt{- x}}{3} \cdot - 3 - 3 \frac{\sqrt{- x}}{3} - 3 \cdot - 3) - 54 (\frac{\sqrt{- x}}{3} - 3) - 81$

Step 5.2.3.3.1.1.5

Add $1$ and $1$ .

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

Step 5.2.3.3.1.1.6

Multiply $3$ by $3$ .

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

Step 5.2.3.3.1.2

Rewrite ${\sqrt{- x}}^{2}$ as $- x$ .

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Step 5.2.3.3.1.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{- x}$ as ${(- x)}^{\frac{1}{2}}$ .

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

Step 5.2.3.3.1.2.2

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

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

Step 5.2.3.3.1.2.3

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

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

Step 5.2.3.3.1.2.4

Cancel the common factor of $2$ .

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Step 5.2.3.3.1.2.4.1

Cancel the common factor.

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

Step 5.2.3.3.1.2.4.2

Rewrite the expression.

$v^{- 1} (\frac{\sqrt{- x}}{3} - 3) = - 9 (\frac{- x}{9} + \frac{\sqrt{- x}}{3} \cdot - 3 - 3 \frac{\sqrt{- x}}{3} - 3 \cdot - 3) - 54 (\frac{\sqrt{- x}}{3} - 3) - 81$

Step 5.2.3.3.1.2.5

Simplify.

$v^{- 1} (\frac{\sqrt{- x}}{3} - 3) = - 9 (\frac{- x}{9} + \frac{\sqrt{- x}}{3} \cdot - 3 - 3 \frac{\sqrt{- x}}{3} - 3 \cdot - 3) - 54 (\frac{\sqrt{- x}}{3} - 3) - 81$

Step 5.2.3.3.1.3

Move the negative in front of the fraction.

$v^{- 1} (\frac{\sqrt{- x}}{3} - 3) = - 9 (- \frac{x}{9} + \frac{\sqrt{- x}}{3} \cdot - 3 - 3 \frac{\sqrt{- x}}{3} - 3 \cdot - 3) - 54 (\frac{\sqrt{- x}}{3} - 3) - 81$

Step 5.2.3.3.1.4

Cancel the common factor of $3$ .

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Step 5.2.3.3.1.4.1

Factor $3$ out of $- 3$ .

$v^{- 1} (\frac{\sqrt{- x}}{3} - 3) = - 9 (- \frac{x}{9} + \frac{\sqrt{- x}}{3} \cdot (3 (- 1)) - 3 \frac{\sqrt{- x}}{3} - 3 \cdot - 3) - 54 (\frac{\sqrt{- x}}{3} - 3) - 81$

Step 5.2.3.3.1.4.2

Cancel the common factor.

$v^{- 1} (\frac{\sqrt{- x}}{3} - 3) = - 9 (- \frac{x}{9} + \frac{\sqrt{- x}}{3} \cdot (3 \cdot - 1) - 3 \frac{\sqrt{- x}}{3} - 3 \cdot - 3) - 54 (\frac{\sqrt{- x}}{3} - 3) - 81$

Step 5.2.3.3.1.4.3

Rewrite the expression.

$v^{- 1} (\frac{\sqrt{- x}}{3} - 3) = - 9 (- \frac{x}{9} + \sqrt{- x} \cdot - 1 - 3 \frac{\sqrt{- x}}{3} - 3 \cdot - 3) - 54 (\frac{\sqrt{- x}}{3} - 3) - 81$

Step 5.2.3.3.1.5

Move $- 1$ to the left of $\sqrt{- x}$ .

$v^{- 1} (\frac{\sqrt{- x}}{3} - 3) = - 9 (- \frac{x}{9} - 1 \cdot \sqrt{- x} - 3 \frac{\sqrt{- x}}{3} - 3 \cdot - 3) - 54 (\frac{\sqrt{- x}}{3} - 3) - 81$

Step 5.2.3.3.1.6

Rewrite $- 1 \sqrt{- x}$ as $- \sqrt{- x}$ .

$v^{- 1} (\frac{\sqrt{- x}}{3} - 3) = - 9 (- \frac{x}{9} - \sqrt{- x} - 3 \frac{\sqrt{- x}}{3} - 3 \cdot - 3) - 54 (\frac{\sqrt{- x}}{3} - 3) - 81$

Step 5.2.3.3.1.7

Cancel the common factor of $3$ .

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Step 5.2.3.3.1.7.1

Factor $3$ out of $- 3$ .

$v^{- 1} (\frac{\sqrt{- x}}{3} - 3) = - 9 (- \frac{x}{9} - \sqrt{- x} + 3 (- 1) (\frac{\sqrt{- x}}{3}) - 3 \cdot - 3) - 54 (\frac{\sqrt{- x}}{3} - 3) - 81$

Step 5.2.3.3.1.7.2

Cancel the common factor.

$v^{- 1} (\frac{\sqrt{- x}}{3} - 3) = - 9 (- \frac{x}{9} - \sqrt{- x} + 3 \cdot (- 1 \frac{\sqrt{- x}}{3}) - 3 \cdot - 3) - 54 (\frac{\sqrt{- x}}{3} - 3) - 81$

Step 5.2.3.3.1.7.3

Rewrite the expression.

$v^{- 1} (\frac{\sqrt{- x}}{3} - 3) = - 9 (- \frac{x}{9} - \sqrt{- x} - 1 \sqrt{- x} - 3 \cdot - 3) - 54 (\frac{\sqrt{- x}}{3} - 3) - 81$

Step 5.2.3.3.1.8

Rewrite $- 1 \sqrt{- x}$ as $- \sqrt{- x}$ .

$v^{- 1} (\frac{\sqrt{- x}}{3} - 3) = - 9 (- \frac{x}{9} - \sqrt{- x} - \sqrt{- x} - 3 \cdot - 3) - 54 (\frac{\sqrt{- x}}{3} - 3) - 81$

Step 5.2.3.3.1.9

Multiply $- 3$ by $- 3$ .

$v^{- 1} (\frac{\sqrt{- x}}{3} - 3) = - 9 (- \frac{x}{9} - \sqrt{- x} - \sqrt{- x} + 9) - 54 (\frac{\sqrt{- x}}{3} - 3) - 81$

Step 5.2.3.3.2

Subtract $\sqrt{- x}$ from $- \sqrt{- x}$ .

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

Step 5.2.3.4

Apply the distributive property.

$v^{- 1} (\frac{\sqrt{- x}}{3} - 3) = - 9 (- \frac{x}{9}) - 9 (- 2 \sqrt{- x}) - 9 \cdot 9 - 54 (\frac{\sqrt{- x}}{3} - 3) - 81$

Step 5.2.3.5

Simplify.

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Step 5.2.3.5.1

Cancel the common factor of $9$ .

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Step 5.2.3.5.1.1

Move the leading negative in $- \frac{x}{9}$ into the numerator.

$v^{- 1} (\frac{\sqrt{- x}}{3} - 3) = - 9 \frac{- x}{9} - 9 (- 2 \sqrt{- x}) - 9 \cdot 9 - 54 (\frac{\sqrt{- x}}{3} - 3) - 81$

Step 5.2.3.5.1.2

Factor $9$ out of $- 9$ .

$v^{- 1} (\frac{\sqrt{- x}}{3} - 3) = 9 (- 1) (\frac{- x}{9}) - 9 (- 2 \sqrt{- x}) - 9 \cdot 9 - 54 (\frac{\sqrt{- x}}{3} - 3) - 81$

Step 5.2.3.5.1.3

Cancel the common factor.

$v^{- 1} (\frac{\sqrt{- x}}{3} - 3) = 9 \cdot (- 1 \frac{- x}{9}) - 9 (- 2 \sqrt{- x}) - 9 \cdot 9 - 54 (\frac{\sqrt{- x}}{3} - 3) - 81$

Step 5.2.3.5.1.4

Rewrite the expression.

$v^{- 1} (\frac{\sqrt{- x}}{3} - 3) = - 1 (- x) - 9 (- 2 \sqrt{- x}) - 9 \cdot 9 - 54 (\frac{\sqrt{- x}}{3} - 3) - 81$

Step 5.2.3.5.2

Multiply $- 1$ by $- 1$ .

$v^{- 1} (\frac{\sqrt{- x}}{3} - 3) = 1 x - 9 (- 2 \sqrt{- x}) - 9 \cdot 9 - 54 (\frac{\sqrt{- x}}{3} - 3) - 81$

Step 5.2.3.5.3

Multiply $x$ by $1$ .

$v^{- 1} (\frac{\sqrt{- x}}{3} - 3) = x - 9 (- 2 \sqrt{- x}) - 9 \cdot 9 - 54 (\frac{\sqrt{- x}}{3} - 3) - 81$

Step 5.2.3.5.4

Multiply $- 2$ by $- 9$ .

$v^{- 1} (\frac{\sqrt{- x}}{3} - 3) = x + 18 \sqrt{- x} - 9 \cdot 9 - 54 (\frac{\sqrt{- x}}{3} - 3) - 81$

Step 5.2.3.5.5

Multiply $- 9$ by $9$ .

$v^{- 1} (\frac{\sqrt{- x}}{3} - 3) = x + 18 \sqrt{- x} - 81 - 54 (\frac{\sqrt{- x}}{3} - 3) - 81$

Step 5.2.3.6

Apply the distributive property.

$v^{- 1} (\frac{\sqrt{- x}}{3} - 3) = x + 18 \sqrt{- x} - 81 - 54 \frac{\sqrt{- x}}{3} - 54 \cdot - 3 - 81$

Step 5.2.3.7

Cancel the common factor of $3$ .

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Step 5.2.3.7.1

Factor $3$ out of $- 54$ .

$v^{- 1} (\frac{\sqrt{- x}}{3} - 3) = x + 18 \sqrt{- x} - 81 + 3 (- 18) (\frac{\sqrt{- x}}{3}) - 54 \cdot - 3 - 81$

Step 5.2.3.7.2

Cancel the common factor.

$v^{- 1} (\frac{\sqrt{- x}}{3} - 3) = x + 18 \sqrt{- x} - 81 + 3 \cdot (- 18 \frac{\sqrt{- x}}{3}) - 54 \cdot - 3 - 81$

Step 5.2.3.7.3

Rewrite the expression.

$v^{- 1} (\frac{\sqrt{- x}}{3} - 3) = x + 18 \sqrt{- x} - 81 - 18 \sqrt{- x} - 54 \cdot - 3 - 81$

Step 5.2.3.8

Multiply $- 54$ by $- 3$ .

$v^{- 1} (\frac{\sqrt{- x}}{3} - 3) = x + 18 \sqrt{- x} - 81 - 18 \sqrt{- x} + 162 - 81$

Step 5.2.4

Simplify by adding terms.

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Step 5.2.4.1

Combine the opposite terms in $x + 18 \sqrt{- x} - 81 - 18 \sqrt{- x} + 162 - 81$ .

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Step 5.2.4.1.1

Subtract $18 \sqrt{- x}$ from $18 \sqrt{- x}$ .

$v^{- 1} (\frac{\sqrt{- x}}{3} - 3) = x + 0 - 81 + 162 - 81$

Step 5.2.4.1.2

Add $x$ and $0$ .

$v^{- 1} (\frac{\sqrt{- x}}{3} - 3) = x - 81 + 162 - 81$

Step 5.2.4.2

Add $- 81$ and $162$ .

$v^{- 1} (\frac{\sqrt{- x}}{3} - 3) = x + 81 - 81$

Step 5.2.4.3

Combine the opposite terms in $x + 81 - 81$ .

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Step 5.2.4.3.1

Subtract $81$ from $81$ .

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

Step 5.2.4.3.2

Add $x$ and $0$ .

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

Step 5.3

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

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Step 5.3.1

Set up the composite result function.

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

Step 5.3.2

Evaluate $v (- 9 x^{2} - 54 x - 81)$ by substituting in the value of $v^{- 1}$ into $v$ .

$v (- 9 x^{2} - 54 x - 81) = \frac{\sqrt{- (- 9 x^{2} - 54 x - 81)}}{3} - 3$

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 $9$ out of $- 9 x^{2} - 54 x - 81$ .

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Step 5.3.3.1.1.1

Factor $9$ out of $- 9 x^{2}$ .

$v (- 9 x^{2} - 54 x - 81) = \frac{\sqrt{- (9 (- x^{2}) - 54 x - 81)}}{3} - 3$

Step 5.3.3.1.1.2

Factor $9$ out of $- 54 x$ .

$v (- 9 x^{2} - 54 x - 81) = \frac{\sqrt{- (9 (- x^{2}) + 9 (- 6 x) - 81)}}{3} - 3$

Step 5.3.3.1.1.3

Factor $9$ out of $- 81$ .

$v (- 9 x^{2} - 54 x - 81) = \frac{\sqrt{- (9 (- x^{2}) + 9 (- 6 x) + 9 (- 9))}}{3} - 3$

Step 5.3.3.1.1.4

Factor $9$ out of $9 (- x^{2}) + 9 (- 6 x)$ .

$v (- 9 x^{2} - 54 x - 81) = \frac{\sqrt{- (9 (- x^{2} - 6 x) + 9 (- 9))}}{3} - 3$

Step 5.3.3.1.1.5

Factor $9$ out of $9 (- x^{2} - 6 x) + 9 (- 9)$ .

$v (- 9 x^{2} - 54 x - 81) = \frac{\sqrt{- (9 (- x^{2} - 6 x - 9))}}{3} - 3$

Step 5.3.3.1.2

Factor by grouping.

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Step 5.3.3.1.2.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = - 1 \cdot - 9 = 9$ and whose sum is $b = - 6$ .

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Step 5.3.3.1.2.1.1

Factor $- 6$ out of $- 6 x$ .

$v (- 9 x^{2} - 54 x - 81) = \frac{\sqrt{- (9 (- x^{2} - 6 x - 9))}}{3} - 3$

Step 5.3.3.1.2.1.2

Rewrite $- 6$ as $- 3$ plus $- 3$

$v (- 9 x^{2} - 54 x - 81) = \frac{\sqrt{- (9 (- x^{2} + (- 3 - 3) x - 9))}}{3} - 3$

Step 5.3.3.1.2.1.3

Apply the distributive property.

$v (- 9 x^{2} - 54 x - 81) = \frac{\sqrt{- (9 (- x^{2} - 3 x - 3 x - 9))}}{3} - 3$

Step 5.3.3.1.2.2

Factor out the greatest common factor from each group.

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Step 5.3.3.1.2.2.1

Group the first two terms and the last two terms.

$v (- 9 x^{2} - 54 x - 81) = \frac{\sqrt{- (9 ((- x^{2} - 3 x) - 3 x - 9))}}{3} - 3$

Step 5.3.3.1.2.2.2

Factor out the greatest common factor (GCF) from each group.

$v (- 9 x^{2} - 54 x - 81) = \frac{\sqrt{- (9 (x (- x - 3) + 3 (- x - 3)))}}{3} - 3$

Step 5.3.3.1.2.3

Factor the polynomial by factoring out the greatest common factor, $- x - 3$ .

$v (- 9 x^{2} - 54 x - 81) = \frac{\sqrt{- (9 ((- x - 3) (x + 3)))}}{3} - 3$

$v (- 9 x^{2} - 54 x - 81) = \frac{\sqrt{- (9 (- x - 3) (x + 3))}}{3} - 3$

Step 5.3.3.1.3

Combine exponents.

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Step 5.3.3.1.3.1

Factor $- 1$ out of $- x$ .

$v (- 9 x^{2} - 54 x - 81) = \frac{\sqrt{- (9 (- (x) - 3) (x + 3))}}{3} - 3$

Step 5.3.3.1.3.2

Rewrite $- 3$ as $- 1 (3)$ .

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

Step 5.3.3.1.3.3

Factor $- 1$ out of $- (x) - 1 (3)$ .

$v (- 9 x^{2} - 54 x - 81) = \frac{\sqrt{- (9 (- (x + 3)) (x + 3))}}{3} - 3$

Step 5.3.3.1.3.4

Rewrite $- (x + 3)$ as $- 1 (x + 3)$ .

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

Step 5.3.3.1.3.5

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

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

Step 5.3.3.1.3.6

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

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

Step 5.3.3.1.3.7

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

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

Step 5.3.3.1.3.8

Add $1$ and $1$ .

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

Step 5.3.3.1.3.9

Multiply $9$ by $- 1$ .

$v (- 9 x^{2} - 54 x - 81) = \frac{\sqrt{- (- 9 {(x + 3)}^{2})}}{3} - 3$

$v (- 9 x^{2} - 54 x - 81) = \frac{\sqrt{9 {(x + 3)}^{2}}}{3} - 3$

Step 5.3.3.1.4

Multiply $- 1$ by $- 9$ .

$v (- 9 x^{2} - 54 x - 81) = \frac{\sqrt{9 {(x + 3)}^{2}}}{3} - 3$

Step 5.3.3.1.5

Rewrite $9 {(x + 3)}^{2}$ as ${(3 (x + 3))}^{2}$ .

$v (- 9 x^{2} - 54 x - 81) = \frac{\sqrt{{(3 (x + 3))}^{2}}}{3} - 3$

Step 5.3.3.1.6

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

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

Step 5.3.3.1.7

Apply the distributive property.

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

Step 5.3.3.1.8

Multiply $3$ by $3$ .

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

Step 5.3.3.1.9

Factor $3$ out of $3 x + 9$ .

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Step 5.3.3.1.9.1

Factor $3$ out of $3 x$ .

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

Step 5.3.3.1.9.2

Factor $3$ out of $9$ .

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

Step 5.3.3.1.9.3

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

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

Step 5.3.3.2

Cancel the common factor of $3$ .

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Step 5.3.3.2.1

Cancel the common factor.

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

Step 5.3.3.2.2

Divide $x + 3$ by $1$ .

$v (- 9 x^{2} - 54 x - 81) = x + 3 - 3$

Step 5.3.4

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

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Step 5.3.4.1

Subtract $3$ from $3$ .

$v (- 9 x^{2} - 54 x - 81) = x + 0$

Step 5.3.4.2

Add $x$ and $0$ .

$v (- 9 x^{2} - 54 x - 81) = x$

Step 5.4

Since $v^{- 1} (v (x)) = x$ and $v (v^{- 1} (x)) = x$ , then $v^{- 1} (x) = - 9 x^{2} - 54 x - 81$ is the inverse of $v (x) = \frac{\sqrt{- x}}{3} - 3$ .

$v^{- 1} (x) = - 9 x^{2} - 54 x - 81$