Trigonometry Examples

$y = 2 + \sqrt{6 x - 5}$

Step 1

Interchange the variables.

$x = 2 + \sqrt{6 y - 5}$

Step 2

Solve for $y$ .

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Step 2.1

Rewrite the equation as $2 + \sqrt{6 y - 5} = x$ .

$2 + \sqrt{6 y - 5} = x$

Step 2.2

Subtract $2$ from both sides of the equation.

$\sqrt{6 y - 5} = x - 2$

Step 2.3

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

${\sqrt{6 y - 5}}^{2} = {(x - 2)}^{2}$

Step 2.4

Simplify each side of the equation.

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Step 2.4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{6 y - 5}$ as ${(6 y - 5)}^{\frac{1}{2}}$ .

${({(6 y - 5)}^{\frac{1}{2}})}^{2} = {(x - 2)}^{2}$

Step 2.4.2

Simplify the left side.

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Step 2.4.2.1

Simplify ${({(6 y - 5)}^{\frac{1}{2}})}^{2}$ .

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Step 2.4.2.1.1

Multiply the exponents in ${({(6 y - 5)}^{\frac{1}{2}})}^{2}$ .

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Step 2.4.2.1.1.1

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

${(6 y - 5)}^{\frac{1}{2} \cdot 2} = {(x - 2)}^{2}$

Step 2.4.2.1.1.2

Cancel the common factor of $2$ .

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Step 2.4.2.1.1.2.1

Cancel the common factor.

${(6 y - 5)}^{\frac{1}{2} \cdot 2} = {(x - 2)}^{2}$

Step 2.4.2.1.1.2.2

Rewrite the expression.

${(6 y - 5)}^{1} = {(x - 2)}^{2}$

Step 2.4.2.1.2

Simplify.

$6 y - 5 = {(x - 2)}^{2}$

Step 2.4.3

Simplify the right side.

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Step 2.4.3.1

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

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Step 2.4.3.1.1

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

$6 y - 5 = (x - 2) (x - 2)$

Step 2.4.3.1.2

Expand $(x - 2) (x - 2)$ using the FOIL Method.

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Step 2.4.3.1.2.1

Apply the distributive property.

$6 y - 5 = x (x - 2) - 2 (x - 2)$

Step 2.4.3.1.2.2

Apply the distributive property.

$6 y - 5 = x \cdot x + x \cdot - 2 - 2 (x - 2)$

Step 2.4.3.1.2.3

Apply the distributive property.

$6 y - 5 = x \cdot x + x \cdot - 2 - 2 x - 2 \cdot - 2$

Step 2.4.3.1.3

Simplify and combine like terms.

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Step 2.4.3.1.3.1

Simplify each term.

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Step 2.4.3.1.3.1.1

Multiply $x$ by $x$ .

$6 y - 5 = x^{2} + x \cdot - 2 - 2 x - 2 \cdot - 2$

Step 2.4.3.1.3.1.2

Move $- 2$ to the left of $x$ .

$6 y - 5 = x^{2} - 2 \cdot x - 2 x - 2 \cdot - 2$

Step 2.4.3.1.3.1.3

Multiply $- 2$ by $- 2$ .

$6 y - 5 = x^{2} - 2 x - 2 x + 4$

Step 2.4.3.1.3.2

Subtract $2 x$ from $- 2 x$ .

$6 y - 5 = x^{2} - 4 x + 4$

Step 2.5

Solve for $y$ .

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Step 2.5.1

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

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Step 2.5.1.1

Add $5$ to both sides of the equation.

$6 y = x^{2} - 4 x + 4 + 5$

Step 2.5.1.2

Add $4$ and $5$ .

$6 y = x^{2} - 4 x + 9$

Step 2.5.2

Divide each term in $6 y = x^{2} - 4 x + 9$ by $6$ and simplify.

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Step 2.5.2.1

Divide each term in $6 y = x^{2} - 4 x + 9$ by $6$ .

$\frac{6 y}{6} = \frac{x^{2}}{6} + \frac{- 4 x}{6} + \frac{9}{6}$

Step 2.5.2.2

Simplify the left side.

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Step 2.5.2.2.1

Cancel the common factor of $6$ .

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Step 2.5.2.2.1.1

Cancel the common factor.

$\frac{6 y}{6} = \frac{x^{2}}{6} + \frac{- 4 x}{6} + \frac{9}{6}$

Step 2.5.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{x^{2}}{6} + \frac{- 4 x}{6} + \frac{9}{6}$

Step 2.5.2.3

Simplify the right side.

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Step 2.5.2.3.1

Simplify each term.

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Step 2.5.2.3.1.1

Cancel the common factor of $- 4$ and $6$ .

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Step 2.5.2.3.1.1.1

Factor $2$ out of $- 4 x$ .

$y = \frac{x^{2}}{6} + \frac{2 (- 2 x)}{6} + \frac{9}{6}$

Step 2.5.2.3.1.1.2

Cancel the common factors.

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Step 2.5.2.3.1.1.2.1

Factor $2$ out of $6$ .

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

Step 2.5.2.3.1.1.2.2

Cancel the common factor.

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

Step 2.5.2.3.1.1.2.3

Rewrite the expression.

$y = \frac{x^{2}}{6} + \frac{- 2 x}{3} + \frac{9}{6}$

Step 2.5.2.3.1.2

Move the negative in front of the fraction.

$y = \frac{x^{2}}{6} - \frac{2 x}{3} + \frac{9}{6}$

Step 2.5.2.3.1.3

Cancel the common factor of $9$ and $6$ .

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Step 2.5.2.3.1.3.1

Factor $3$ out of $9$ .

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

Step 2.5.2.3.1.3.2

Cancel the common factors.

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Step 2.5.2.3.1.3.2.1

Factor $3$ out of $6$ .

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

Step 2.5.2.3.1.3.2.2

Cancel the common factor.

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

Step 2.5.2.3.1.3.2.3

Rewrite the expression.

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

Step 3

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

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

Step 4

Verify if $f^{- 1} (x) = \frac{x^{2}}{6} - \frac{2 x}{3} + \frac{3}{2}$ is the inverse of $f (x) = 2 + \sqrt{6 x - 5}$ .

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Step 4.1

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

Step 4.2

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

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Step 4.2.1

Set up the composite result function.

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

Step 4.2.2

Evaluate $f^{- 1} (2 + \sqrt{6 x - 5})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (2 + \sqrt{6 x - 5}) = \frac{{(2 + \sqrt{6 x - 5})}^{2}}{6} - \frac{2 (2 + \sqrt{6 x - 5})}{3} + \frac{3}{2}$

Step 4.2.3

Simplify each term.

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Step 4.2.3.1

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

$f^{- 1} (2 + \sqrt{6 x - 5}) = \frac{(2 + \sqrt{6 x - 5}) (2 + \sqrt{6 x - 5})}{6} - \frac{2 (2 + \sqrt{6 x - 5})}{3} + \frac{3}{2}$

Step 4.2.3.2

Expand $(2 + \sqrt{6 x - 5}) (2 + \sqrt{6 x - 5})$ using the FOIL Method.

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Step 4.2.3.2.1

Apply the distributive property.

$f^{- 1} (2 + \sqrt{6 x - 5}) = \frac{2 (2 + \sqrt{6 x - 5}) + \sqrt{6 x - 5} (2 + \sqrt{6 x - 5})}{6} - \frac{2 (2 + \sqrt{6 x - 5})}{3} + \frac{3}{2}$

Step 4.2.3.2.2

Apply the distributive property.

$f^{- 1} (2 + \sqrt{6 x - 5}) = \frac{2 \cdot 2 + 2 \sqrt{6 x - 5} + \sqrt{6 x - 5} (2 + \sqrt{6 x - 5})}{6} - \frac{2 (2 + \sqrt{6 x - 5})}{3} + \frac{3}{2}$

Step 4.2.3.2.3

Apply the distributive property.

$f^{- 1} (2 + \sqrt{6 x - 5}) = \frac{2 \cdot 2 + 2 \sqrt{6 x - 5} + \sqrt{6 x - 5} \cdot 2 + \sqrt{6 x - 5} \sqrt{6 x - 5}}{6} - \frac{2 (2 + \sqrt{6 x - 5})}{3} + \frac{3}{2}$

Step 4.2.3.3

Simplify and combine like terms.

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Step 4.2.3.3.1

Simplify each term.

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Step 4.2.3.3.1.1

Multiply $2$ by $2$ .

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

Step 4.2.3.3.1.2

Move $2$ to the left of $\sqrt{6 x - 5}$ .

$f^{- 1} (2 + \sqrt{6 x - 5}) = \frac{4 + 2 \sqrt{6 x - 5} + 2 \sqrt{6 x - 5} + \sqrt{6 x - 5} \sqrt{6 x - 5}}{6} - \frac{2 (2 + \sqrt{6 x - 5})}{3} + \frac{3}{2}$

Step 4.2.3.3.1.3

Multiply $\sqrt{6 x - 5} \sqrt{6 x - 5}$ .

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Step 4.2.3.3.1.3.1

Raise $\sqrt{6 x - 5}$ to the power of $1$ .

$f^{- 1} (2 + \sqrt{6 x - 5}) = \frac{4 + 2 \sqrt{6 x - 5} + 2 \sqrt{6 x - 5} + \sqrt{6 x - 5} \sqrt{6 x - 5}}{6} - \frac{2 (2 + \sqrt{6 x - 5})}{3} + \frac{3}{2}$

Step 4.2.3.3.1.3.2

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

$f^{- 1} (2 + \sqrt{6 x - 5}) = \frac{4 + 2 \sqrt{6 x - 5} + 2 \sqrt{6 x - 5} + {\sqrt{6 x - 5}}^{1 + 1}}{6} - \frac{2 (2 + \sqrt{6 x - 5})}{3} + \frac{3}{2}$

Step 4.2.3.3.1.3.3

Add $1$ and $1$ .

$f^{- 1} (2 + \sqrt{6 x - 5}) = \frac{4 + 2 \sqrt{6 x - 5} + 2 \sqrt{6 x - 5} + {\sqrt{6 x - 5}}^{2}}{6} - \frac{2 (2 + \sqrt{6 x - 5})}{3} + \frac{3}{2}$

Step 4.2.3.3.1.4

Rewrite ${\sqrt{6 x - 5}}^{2}$ as $6 x - 5$ .

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Step 4.2.3.3.1.4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{6 x - 5}$ as ${(6 x - 5)}^{\frac{1}{2}}$ .

$f^{- 1} (2 + \sqrt{6 x - 5}) = \frac{4 + 2 \sqrt{6 x - 5} + 2 \sqrt{6 x - 5} + {({(6 x - 5)}^{\frac{1}{2}})}^{2}}{6} - \frac{2 (2 + \sqrt{6 x - 5})}{3} + \frac{3}{2}$

Step 4.2.3.3.1.4.2

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

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

Step 4.2.3.3.1.4.3

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

$f^{- 1} (2 + \sqrt{6 x - 5}) = \frac{4 + 2 \sqrt{6 x - 5} + 2 \sqrt{6 x - 5} + {(6 x - 5)}^{\frac{2}{2}}}{6} - \frac{2 (2 + \sqrt{6 x - 5})}{3} + \frac{3}{2}$

Step 4.2.3.3.1.4.4

Cancel the common factor of $2$ .

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Step 4.2.3.3.1.4.4.1

Cancel the common factor.

$f^{- 1} (2 + \sqrt{6 x - 5}) = \frac{4 + 2 \sqrt{6 x - 5} + 2 \sqrt{6 x - 5} + {(6 x - 5)}^{\frac{2}{2}}}{6} - \frac{2 (2 + \sqrt{6 x - 5})}{3} + \frac{3}{2}$

Step 4.2.3.3.1.4.4.2

Rewrite the expression.

$f^{- 1} (2 + \sqrt{6 x - 5}) = \frac{4 + 2 \sqrt{6 x - 5} + 2 \sqrt{6 x - 5} + 6 x - 5}{6} - \frac{2 (2 + \sqrt{6 x - 5})}{3} + \frac{3}{2}$

Step 4.2.3.3.1.4.5

Simplify.

$f^{- 1} (2 + \sqrt{6 x - 5}) = \frac{4 + 2 \sqrt{6 x - 5} + 2 \sqrt{6 x - 5} + 6 x - 5}{6} - \frac{2 (2 + \sqrt{6 x - 5})}{3} + \frac{3}{2}$

Step 4.2.3.3.2

Subtract $5$ from $4$ .

$f^{- 1} (2 + \sqrt{6 x - 5}) = \frac{- 1 + 2 \sqrt{6 x - 5} + 2 \sqrt{6 x - 5} + 6 x}{6} - \frac{2 (2 + \sqrt{6 x - 5})}{3} + \frac{3}{2}$

Step 4.2.3.3.3

Add $2 \sqrt{6 x - 5}$ and $2 \sqrt{6 x - 5}$ .

$f^{- 1} (2 + \sqrt{6 x - 5}) = \frac{- 1 + 4 \sqrt{6 x - 5} + 6 x}{6} - \frac{2 (2 + \sqrt{6 x - 5})}{3} + \frac{3}{2}$

Step 4.2.4

To write $- \frac{2 (2 + \sqrt{6 x - 5})}{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 4.2.5

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.2.5.1

Multiply $\frac{2 (2 + \sqrt{6 x - 5})}{3}$ by $\frac{2}{2}$ .

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

Step 4.2.5.2

Multiply $3$ by $2$ .

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

Step 4.2.6

Combine the numerators over the common denominator.

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

Step 4.2.7

Simplify each term.

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Step 4.2.7.1

Simplify the numerator.

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Step 4.2.7.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{6 x - 5}$ as ${(6 x - 5)}^{\frac{1}{2}}$ .

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

Step 4.2.7.1.2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{6 x - 5}$ as ${(6 x - 5)}^{\frac{1}{2}}$ .

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

Step 4.2.7.1.3

Apply the distributive property.

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

Step 4.2.7.1.4

Multiply $- 2$ by $2$ .

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

Step 4.2.7.1.5

Apply the distributive property.

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

Step 4.2.7.1.6

Multiply $- 4$ by $2$ .

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

Step 4.2.7.1.7

Multiply $2$ by $- 2$ .

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

Step 4.2.7.1.8

Subtract $8$ from $- 1$ .

$f^{- 1} (2 + \sqrt{6 x - 5}) = \frac{4 {(6 x - 5)}^{\frac{1}{2}} + 6 x - 9 - 4 {(6 x - 5)}^{\frac{1}{2}}}{6} + \frac{3}{2}$

Step 4.2.7.1.9

Subtract $4 {(6 x - 5)}^{\frac{1}{2}}$ from $4 {(6 x - 5)}^{\frac{1}{2}}$ .

$f^{- 1} (2 + \sqrt{6 x - 5}) = \frac{6 x - 9 + 0}{6} + \frac{3}{2}$

Step 4.2.7.1.10

Add $6 x - 9$ and $0$ .

$f^{- 1} (2 + \sqrt{6 x - 5}) = \frac{6 x - 9}{6} + \frac{3}{2}$

Step 4.2.7.1.11

Factor $3$ out of $6 x - 9$ .

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Step 4.2.7.1.11.1

Factor $3$ out of $6 x$ .

$f^{- 1} (2 + \sqrt{6 x - 5}) = \frac{3 (2 x) - 9}{6} + \frac{3}{2}$

Step 4.2.7.1.11.2

Factor $3$ out of $- 9$ .

$f^{- 1} (2 + \sqrt{6 x - 5}) = \frac{3 (2 x) + 3 (- 3)}{6} + \frac{3}{2}$

Step 4.2.7.1.11.3

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

$f^{- 1} (2 + \sqrt{6 x - 5}) = \frac{3 (2 x - 3)}{6} + \frac{3}{2}$

Step 4.2.7.2

Cancel the common factors.

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Step 4.2.7.2.1

Factor $3$ out of $6$ .

$f^{- 1} (2 + \sqrt{6 x - 5}) = \frac{3 (2 x - 3)}{3 \cdot 2} + \frac{3}{2}$

Step 4.2.7.2.2

Cancel the common factor.

$f^{- 1} (2 + \sqrt{6 x - 5}) = \frac{3 (2 x - 3)}{3 \cdot 2} + \frac{3}{2}$

Step 4.2.7.2.3

Rewrite the expression.

$f^{- 1} (2 + \sqrt{6 x - 5}) = \frac{2 x - 3}{2} + \frac{3}{2}$

Step 4.2.8

Simplify terms.

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Step 4.2.8.1

Combine the numerators over the common denominator.

$f^{- 1} (2 + \sqrt{6 x - 5}) = \frac{2 x - 3 + 3}{2}$

Step 4.2.8.2

Combine the opposite terms in $2 x - 3 + 3$ .

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Step 4.2.8.2.1

Add $- 3$ and $3$ .

$f^{- 1} (2 + \sqrt{6 x - 5}) = \frac{2 x + 0}{2}$

Step 4.2.8.2.2

Add $2 x$ and $0$ .

$f^{- 1} (2 + \sqrt{6 x - 5}) = \frac{2 x}{2}$

Step 4.2.8.3

Cancel the common factor of $2$ .

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Step 4.2.8.3.1

Cancel the common factor.

$f^{- 1} (2 + \sqrt{6 x - 5}) = \frac{2 x}{2}$

Step 4.2.8.3.2

Divide $x$ by $1$ .

$f^{- 1} (2 + \sqrt{6 x - 5}) = x$

Step 4.3

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

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Step 4.3.1

Set up the composite result function.

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

Step 4.3.2

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

$f (\frac{x^{2}}{6} - \frac{2 x}{3} + \frac{3}{2}) = 2 + \sqrt{6 (\frac{x^{2}}{6} - \frac{2 x}{3} + \frac{3}{2}) - 5}$

Step 4.3.3

Simplify each term.

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Step 4.3.3.1

Apply the distributive property.

$f (\frac{x^{2}}{6} - \frac{2 x}{3} + \frac{3}{2}) = 2 + \sqrt{6 (\frac{x^{2}}{6}) + 6 (- \frac{2 x}{3}) + 6 (\frac{3}{2}) - 5}$

Step 4.3.3.2

Simplify.

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Step 4.3.3.2.1

Cancel the common factor of $6$ .

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Step 4.3.3.2.1.1

Cancel the common factor.

$f (\frac{x^{2}}{6} - \frac{2 x}{3} + \frac{3}{2}) = 2 + \sqrt{6 (\frac{x^{2}}{6}) + 6 (- \frac{2 x}{3}) + 6 (\frac{3}{2}) - 5}$

Step 4.3.3.2.1.2

Rewrite the expression.

$f (\frac{x^{2}}{6} - \frac{2 x}{3} + \frac{3}{2}) = 2 + \sqrt{x^{2} + 6 (- \frac{2 x}{3}) + 6 (\frac{3}{2}) - 5}$

Step 4.3.3.2.2

Cancel the common factor of $3$ .

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Step 4.3.3.2.2.1

Move the leading negative in $- \frac{2 x}{3}$ into the numerator.

$f (\frac{x^{2}}{6} - \frac{2 x}{3} + \frac{3}{2}) = 2 + \sqrt{x^{2} + 6 (\frac{- 2 x}{3}) + 6 (\frac{3}{2}) - 5}$

Step 4.3.3.2.2.2

Factor $3$ out of $6$ .

$f (\frac{x^{2}}{6} - \frac{2 x}{3} + \frac{3}{2}) = 2 + \sqrt{x^{2} + 3 (2) (\frac{- 2 x}{3}) + 6 (\frac{3}{2}) - 5}$

Step 4.3.3.2.2.3

Cancel the common factor.

$f (\frac{x^{2}}{6} - \frac{2 x}{3} + \frac{3}{2}) = 2 + \sqrt{x^{2} + 3 \cdot (2 (\frac{- 2 x}{3})) + 6 (\frac{3}{2}) - 5}$

Step 4.3.3.2.2.4

Rewrite the expression.

$f (\frac{x^{2}}{6} - \frac{2 x}{3} + \frac{3}{2}) = 2 + \sqrt{x^{2} + 2 (- 2 x) + 6 (\frac{3}{2}) - 5}$

Step 4.3.3.2.3

Multiply $- 2$ by $2$ .

$f (\frac{x^{2}}{6} - \frac{2 x}{3} + \frac{3}{2}) = 2 + \sqrt{x^{2} - 4 x + 6 (\frac{3}{2}) - 5}$

Step 4.3.3.2.4

Cancel the common factor of $2$ .

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Step 4.3.3.2.4.1

Factor $2$ out of $6$ .

$f (\frac{x^{2}}{6} - \frac{2 x}{3} + \frac{3}{2}) = 2 + \sqrt{x^{2} - 4 x + 2 (3) (\frac{3}{2}) - 5}$

Step 4.3.3.2.4.2

Cancel the common factor.

$f (\frac{x^{2}}{6} - \frac{2 x}{3} + \frac{3}{2}) = 2 + \sqrt{x^{2} - 4 x + 2 \cdot (3 (\frac{3}{2})) - 5}$

Step 4.3.3.2.4.3

Rewrite the expression.

$f (\frac{x^{2}}{6} - \frac{2 x}{3} + \frac{3}{2}) = 2 + \sqrt{x^{2} - 4 x + 3 \cdot 3 - 5}$

Step 4.3.3.2.5

Multiply $3$ by $3$ .

$f (\frac{x^{2}}{6} - \frac{2 x}{3} + \frac{3}{2}) = 2 + \sqrt{x^{2} - 4 x + 9 - 5}$

Step 4.3.3.3

Subtract $5$ from $9$ .

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

Step 4.3.3.4

Factor using the perfect square rule.

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Step 4.3.3.4.1

Rewrite $4$ as $2^{2}$ .

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

Step 4.3.3.4.2

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

$4 x = 2 \cdot x \cdot 2$

Step 4.3.3.4.3

Rewrite the polynomial.

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

Step 4.3.3.4.4

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

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

Step 4.3.3.5

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

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

Step 4.3.4

Combine the opposite terms in $2 + x - 2$ .

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Step 4.3.4.1

Subtract $2$ from $2$ .

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

Step 4.3.4.2

Add $x$ and $0$ .

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

Step 4.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{x^{2}}{6} - \frac{2 x}{3} + \frac{3}{2}$ is the inverse of $f (x) = 2 + \sqrt{6 x - 5}$ .

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