Calculus Examples

$f (x) = 1 + \sqrt{3 + 6 x}$

Step 1

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

$y = 1 + \sqrt{3 + 6 x}$

Step 2

Interchange the variables.

$x = 1 + \sqrt{3 + 6 y}$

Step 3

Solve for $y$ .

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Step 3.1

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

$1 + \sqrt{3 + 6 y} = x$

Step 3.2

Subtract $1$ from both sides of the equation.

$\sqrt{3 + 6 y} = x - 1$

Step 3.3

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

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

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 $\sqrt{3 + 6 y}$ as ${(3 + 6 y)}^{\frac{1}{2}}$ .

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

Step 3.4.2

Simplify the left side.

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Step 3.4.2.1

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

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Step 3.4.2.1.1

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

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Step 3.4.2.1.1.1

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

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

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.

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

Step 3.4.2.1.1.2.2

Rewrite the expression.

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

Step 3.4.2.1.2

Simplify.

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

Step 3.4.3

Simplify the right side.

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Step 3.4.3.1

Simplify ${(x - 1)}^{2}$ .

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Step 3.4.3.1.1

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

$3 + 6 y = (x - 1) (x - 1)$

Step 3.4.3.1.2

Expand $(x - 1) (x - 1)$ using the FOIL Method.

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Step 3.4.3.1.2.1

Apply the distributive property.

$3 + 6 y = x (x - 1) - 1 (x - 1)$

Step 3.4.3.1.2.2

Apply the distributive property.

$3 + 6 y = x \cdot x + x \cdot - 1 - 1 (x - 1)$

Step 3.4.3.1.2.3

Apply the distributive property.

$3 + 6 y = x \cdot x + x \cdot - 1 - 1 x - 1 \cdot - 1$

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

Multiply $x$ by $x$ .

$3 + 6 y = x^{2} + x \cdot - 1 - 1 x - 1 \cdot - 1$

Step 3.4.3.1.3.1.2

Move $- 1$ to the left of $x$ .

$3 + 6 y = x^{2} - 1 \cdot x - 1 x - 1 \cdot - 1$

Step 3.4.3.1.3.1.3

Rewrite $- 1 x$ as $- x$ .

$3 + 6 y = x^{2} - x - 1 x - 1 \cdot - 1$

Step 3.4.3.1.3.1.4

Rewrite $- 1 x$ as $- x$ .

$3 + 6 y = x^{2} - x - x - 1 \cdot - 1$

Step 3.4.3.1.3.1.5

Multiply $- 1$ by $- 1$ .

$3 + 6 y = x^{2} - x - x + 1$

Step 3.4.3.1.3.2

Subtract $x$ from $- x$ .

$3 + 6 y = x^{2} - 2 x + 1$

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

Subtract $3$ from both sides of the equation.

$6 y = x^{2} - 2 x + 1 - 3$

Step 3.5.1.2

Subtract $3$ from $1$ .

$6 y = x^{2} - 2 x - 2$

Step 3.5.2

Divide each term in $6 y = x^{2} - 2 x - 2$ by $6$ and simplify.

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Step 3.5.2.1

Divide each term in $6 y = x^{2} - 2 x - 2$ by $6$ .

$\frac{6 y}{6} = \frac{x^{2}}{6} + \frac{- 2 x}{6} + \frac{- 2}{6}$

Step 3.5.2.2

Simplify the left side.

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Step 3.5.2.2.1

Cancel the common factor of $6$ .

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Step 3.5.2.2.1.1

Cancel the common factor.

$\frac{6 y}{6} = \frac{x^{2}}{6} + \frac{- 2 x}{6} + \frac{- 2}{6}$

Step 3.5.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{x^{2}}{6} + \frac{- 2 x}{6} + \frac{- 2}{6}$

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 $- 2$ and $6$ .

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Step 3.5.2.3.1.1.1

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

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

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 $6$ .

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

Step 3.5.2.3.1.1.2.2

Cancel the common factor.

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

Step 3.5.2.3.1.1.2.3

Rewrite the expression.

$y = \frac{x^{2}}{6} + \frac{- x}{3} + \frac{- 2}{6}$

Step 3.5.2.3.1.2

Move the negative in front of the fraction.

$y = \frac{x^{2}}{6} - \frac{x}{3} + \frac{- 2}{6}$

Step 3.5.2.3.1.3

Cancel the common factor of $- 2$ and $6$ .

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Step 3.5.2.3.1.3.1

Factor $2$ out of $- 2$ .

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

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 $2$ out of $6$ .

$y = \frac{x^{2}}{6} - \frac{x}{3} + \frac{2 \cdot - 1}{2 \cdot 3}$

Step 3.5.2.3.1.3.2.2

Cancel the common factor.

$y = \frac{x^{2}}{6} - \frac{x}{3} + \frac{2 \cdot - 1}{2 \cdot 3}$

Step 3.5.2.3.1.3.2.3

Rewrite the expression.

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

Step 3.5.2.3.1.4

Move the negative in front of the fraction.

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

Step 4

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

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

Step 5

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

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

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

Step 5.2.3

Combine the numerators over the common denominator.

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

Step 5.2.4

Simplify each term.

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Step 5.2.4.1

Factor $3$ out of $3 + 6 x$ .

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Step 5.2.4.1.1

Factor $3$ out of $3$ .

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

Step 5.2.4.1.2

Factor $3$ out of $6 x$ .

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

Step 5.2.4.1.3

Factor $3$ out of $3 (1) + 3 (2 x)$ .

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

Step 5.2.4.2

Apply the distributive property.

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

Step 5.2.4.3

Multiply $- 1$ by $1$ .

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

Step 5.2.5

Simplify the expression.

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Step 5.2.5.1

Subtract $1$ from $- 1$ .

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

Step 5.2.5.2

Reorder $1$ and $2 x$ .

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

Step 5.2.6

Simplify each term.

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Step 5.2.6.1

Factor $3$ out of $3 + 6 x$ .

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Step 5.2.6.1.1

Factor $3$ out of $3$ .

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

Step 5.2.6.1.2

Factor $3$ out of $6 x$ .

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

Step 5.2.6.1.3

Factor $3$ out of $3 (1) + 3 (2 x)$ .

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

Step 5.2.6.2

Rewrite ${(1 + \sqrt{3 (1 + 2 x)})}^{2}$ as $(1 + \sqrt{3 (1 + 2 x)}) (1 + \sqrt{3 (1 + 2 x)})$ .

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

Step 5.2.6.3

Expand $(1 + \sqrt{3 (1 + 2 x)}) (1 + \sqrt{3 (1 + 2 x)})$ using the FOIL Method.

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Step 5.2.6.3.1

Apply the distributive property.

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

Step 5.2.6.3.2

Apply the distributive property.

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

Step 5.2.6.3.3

Apply the distributive property.

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

Step 5.2.6.4

Simplify and combine like terms.

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Step 5.2.6.4.1

Simplify each term.

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Step 5.2.6.4.1.1

Multiply $1$ by $1$ .

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

Step 5.2.6.4.1.2

Multiply $\sqrt{3 (1 + 2 x)}$ by $1$ .

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

Step 5.2.6.4.1.3

Multiply $\sqrt{3 (1 + 2 x)}$ by $1$ .

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

Step 5.2.6.4.1.4

Multiply $\sqrt{3 (1 + 2 x)} \sqrt{3 (1 + 2 x)}$ .

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Step 5.2.6.4.1.4.1

Raise $\sqrt{3 (1 + 2 x)}$ to the power of $1$ .

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

Step 5.2.6.4.1.4.2

Raise $\sqrt{3 (1 + 2 x)}$ to the power of $1$ .

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

Step 5.2.6.4.1.4.3

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

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

Step 5.2.6.4.1.4.4

Add $1$ and $1$ .

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

Step 5.2.6.4.1.5

Rewrite ${\sqrt{3 (1 + 2 x)}}^{2}$ as $3 (1 + 2 x)$ .

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Step 5.2.6.4.1.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3 (1 + 2 x)}$ as ${(3 (1 + 2 x))}^{\frac{1}{2}}$ .

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

Step 5.2.6.4.1.5.2

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

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

Step 5.2.6.4.1.5.3

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

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

Step 5.2.6.4.1.5.4

Cancel the common factor of $2$ .

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Step 5.2.6.4.1.5.4.1

Cancel the common factor.

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

Step 5.2.6.4.1.5.4.2

Rewrite the expression.

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

Step 5.2.6.4.1.5.5

Simplify.

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

Step 5.2.6.4.1.6

Apply the distributive property.

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

Step 5.2.6.4.1.7

Multiply $3$ by $1$ .

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

Step 5.2.6.4.1.8

Multiply $2$ by $3$ .

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

Step 5.2.6.4.2

Add $1$ and $3$ .

$f^{- 1} (1 + \sqrt{3 + 6 x}) = \frac{4 + \sqrt{3 (1 + 2 x)} + \sqrt{3 (1 + 2 x)} + 6 x}{6} + \frac{- 2 - \sqrt{3 (2 x + 1)}}{3}$

Step 5.2.6.4.3

Add $\sqrt{3 (1 + 2 x)}$ and $\sqrt{3 (1 + 2 x)}$ .

$f^{- 1} (1 + \sqrt{3 + 6 x}) = \frac{4 + 2 \sqrt{3 (1 + 2 x)} + 6 x}{6} + \frac{- 2 - \sqrt{3 (2 x + 1)}}{3}$

Step 5.2.6.5

Cancel the common factor of $4 + 2 \sqrt{3 (1 + 2 x)} + 6 x$ and $6$ .

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Step 5.2.6.5.1

Factor $2$ out of $4$ .

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

Step 5.2.6.5.2

Factor $2$ out of $2 \sqrt{3 (1 + 2 x)}$ .

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

Step 5.2.6.5.3

Factor $2$ out of $2 \cdot 2 + 2 (\sqrt{3 (1 + 2 x)})$ .

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

Step 5.2.6.5.4

Factor $2$ out of $6 x$ .

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

Step 5.2.6.5.5

Factor $2$ out of $2 \cdot (2 + \sqrt{3 (1 + 2 x)}) + 2 (3 x)$ .

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

Step 5.2.6.5.6

Cancel the common factors.

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Step 5.2.6.5.6.1

Factor $2$ out of $6$ .

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

Step 5.2.6.5.6.2

Cancel the common factor.

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

Step 5.2.6.5.6.3

Rewrite the expression.

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

Step 5.2.6.6

Factor $- 1$ out of $- 2 - \sqrt{3 (2 x + 1)}$ .

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Step 5.2.6.6.1

Rewrite $- 2$ as $- 1 (2)$ .

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

Step 5.2.6.6.2

Factor $- 1$ out of $- \sqrt{3 (2 x + 1)}$ .

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

Step 5.2.6.6.3

Factor $- 1$ out of $- 1 (2) - (\sqrt{3 (2 x + 1)})$ .

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

Step 5.2.6.6.4

Rewrite $- 1 (2 + \sqrt{3 (2 x + 1)})$ as $- (2 + \sqrt{3 (2 x + 1)})$ .

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

Step 5.2.6.7

Move the negative in front of the fraction.

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

Step 5.2.7

Combine the numerators over the common denominator.

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

Step 5.2.8

Simplify each term.

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Step 5.2.8.1

Apply the distributive property.

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

Step 5.2.8.2

Multiply $- 1$ by $2$ .

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

Step 5.2.9

Combine the opposite terms in $2 + \sqrt{3 (1 + 2 x)} + 3 x - 2 - \sqrt{3 (2 x + 1)}$ .

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Step 5.2.9.1

Subtract $2$ from $2$ .

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

Step 5.2.9.2

Add $0$ and $\sqrt{3 (1 + 2 x)}$ .

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

Step 5.2.10

Subtract $\sqrt{3 (2 x + 1)}$ from $\sqrt{3 (1 + 2 x)}$ .

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Step 5.2.10.1

Reorder $1$ and $2 x$ .

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

Step 5.2.10.2

Subtract $\sqrt{3 (2 x + 1)}$ from $\sqrt{3 (2 x + 1)}$ .

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

Step 5.2.11

Add $3 x$ and $0$ .

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

Step 5.2.12

Cancel the common factor of $3$ .

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Step 5.2.12.1

Cancel the common factor.

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

Step 5.2.12.2

Divide $x$ by $1$ .

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

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

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

Step 5.3.3

Simplify each term.

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Step 5.3.3.1

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

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Step 5.3.3.1.1

Factor $3$ out of $3$ .

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

Step 5.3.3.1.2

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

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

Step 5.3.3.1.3

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

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

Step 5.3.3.2

Apply the distributive property.

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

Step 5.3.3.3

Simplify.

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Step 5.3.3.3.1

Cancel the common factor of $2$ .

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Step 5.3.3.3.1.1

Factor $2$ out of $6$ .

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

Step 5.3.3.3.1.2

Cancel the common factor.

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

Step 5.3.3.3.1.3

Rewrite the expression.

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

Step 5.3.3.3.2

Multiply $2 (- \frac{x}{3})$ .

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Step 5.3.3.3.2.1

Multiply $- 1$ by $2$ .

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

Step 5.3.3.3.2.2

Combine $- 2$ and $\frac{x}{3}$ .

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

Step 5.3.3.3.3

Multiply $2 (- \frac{1}{3})$ .

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Step 5.3.3.3.3.1

Multiply $- 1$ by $2$ .

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

Step 5.3.3.3.3.2

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

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

Step 5.3.3.4

Simplify each term.

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Step 5.3.3.4.1

Move the negative in front of the fraction.

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

Step 5.3.3.4.2

Move the negative in front of the fraction.

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

Step 5.3.3.5

Write $1$ as a fraction with a common denominator.

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

Step 5.3.3.6

Combine the numerators over the common denominator.

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

Step 5.3.3.7

Subtract $2$ from $3$ .

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

Step 5.3.3.8

Combine the numerators over the common denominator.

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

Step 5.3.3.9

Factor using the perfect square rule.

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Step 5.3.3.9.1

Rewrite $1$ as $1^{2}$ .

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

Step 5.3.3.9.2

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

$2 x = 2 \cdot x \cdot 1$

Step 5.3.3.9.3

Rewrite the polynomial.

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

Step 5.3.3.9.4

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = x$ and $b = 1$ .

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

Step 5.3.3.10

Combine $3$ and $\frac{{(x - 1)}^{2}}{3}$ .

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

Step 5.3.3.11

Reduce the expression by cancelling the common factors.

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Step 5.3.3.11.1

Reduce the expression $\frac{3 {(x - 1)}^{2}}{3}$ by cancelling the common factors.

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Step 5.3.3.11.1.1

Cancel the common factor.

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

Step 5.3.3.11.1.2

Rewrite the expression.

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

Step 5.3.3.11.2

Divide ${(x - 1)}^{2}$ by $1$ .

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

Step 5.3.3.12

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

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

Step 5.3.4

Combine the opposite terms in $1 + x - 1$ .

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Step 5.3.4.1

Subtract $1$ from $1$ .

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

Step 5.3.4.2

Add $x$ and $0$ .

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

Step 5.4

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

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