Find the Inverse (x-4)^2-2/3=6y-12

${(x - 4)}^{2} - \frac{2}{3} = 6 y - 12$

Rewrite the equation as $6 y - 12 = {(x - 4)}^{2} - \frac{2}{3}$ .

$6 y - 12 = {(x - 4)}^{2} - \frac{2}{3}$

Simplify ${(x - 4)}^{2} - \frac{2}{3}$ .

Simplify each term.

Rewrite ${(x - 4)}^{2}$ as $(x - 4) (x - 4)$ .

$6 y - 12 = (x - 4) (x - 4) - \frac{2}{3}$

Expand $(x - 4) (x - 4)$ using the FOIL Method.

Apply the distributive property.

$6 y - 12 = x (x - 4) - 4 (x - 4) - \frac{2}{3}$

Apply the distributive property.

$6 y - 12 = x \cdot x + x \cdot - 4 - 4 (x - 4) - \frac{2}{3}$

Apply the distributive property.

$6 y - 12 = x \cdot x + x \cdot - 4 - 4 x - 4 \cdot - 4 - \frac{2}{3}$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$6 y - 12 = x^{2} + x \cdot - 4 - 4 x - 4 \cdot - 4 - \frac{2}{3}$

Move $- 4$ to the left of $x$ .

$6 y - 12 = x^{2} - 4 \cdot x - 4 x - 4 \cdot - 4 - \frac{2}{3}$

Multiply $- 4$ by $- 4$ .

$6 y - 12 = x^{2} - 4 x - 4 x + 16 - \frac{2}{3}$

Subtract $4 x$ from $- 4 x$ .

$6 y - 12 = x^{2} - 8 x + 16 - \frac{2}{3}$

To write $16$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$6 y - 12 = x^{2} - 8 x + 16 \cdot \frac{3}{3} - \frac{2}{3}$

Combine $16$ and $\frac{3}{3}$ .

$6 y - 12 = x^{2} - 8 x + \frac{16 \cdot 3}{3} - \frac{2}{3}$

Combine the numerators over the common denominator.

$6 y - 12 = x^{2} - 8 x + \frac{16 \cdot 3 - 2}{3}$

Simplify the numerator.

Multiply $16$ by $3$ .

$6 y - 12 = x^{2} - 8 x + \frac{48 - 2}{3}$

Subtract $2$ from $48$ .

$6 y - 12 = x^{2} - 8 x + \frac{46}{3}$

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

Add $12$ to both sides of the equation.

$6 y = x^{2} - 8 x + \frac{46}{3} + 12$

To write $12$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$6 y = x^{2} - 8 x + \frac{46}{3} + 12 \cdot \frac{3}{3}$

Combine $12$ and $\frac{3}{3}$ .

$6 y = x^{2} - 8 x + \frac{46}{3} + \frac{12 \cdot 3}{3}$

Combine the numerators over the common denominator.

$6 y = x^{2} - 8 x + \frac{46 + 12 \cdot 3}{3}$

Simplify the numerator.

Multiply $12$ by $3$ .

$6 y = x^{2} - 8 x + \frac{46 + 36}{3}$

Add $46$ and $36$ .

$6 y = x^{2} - 8 x + \frac{82}{3}$

Divide each term in $6 y = x^{2} - 8 x + \frac{82}{3}$ by $6$ and simplify.

Divide each term in $6 y = x^{2} - 8 x + \frac{82}{3}$ by $6$ .

$\frac{6 y}{6} = \frac{x^{2}}{6} + \frac{- 8 x}{6} + \frac{\frac{82}{3}}{6}$

Simplify the left side.

Cancel the common factor of $6$ .

Cancel the common factor.

$\frac{6 y}{6} = \frac{x^{2}}{6} + \frac{- 8 x}{6} + \frac{\frac{82}{3}}{6}$

Divide $y$ by $1$ .

$y = \frac{x^{2}}{6} + \frac{- 8 x}{6} + \frac{\frac{82}{3}}{6}$

Simplify the right side.

Simplify each term.

Cancel the common factor of $- 8$ and $6$ .

Factor $2$ out of $- 8 x$ .

$y = \frac{x^{2}}{6} + \frac{2 (- 4 x)}{6} + \frac{\frac{82}{3}}{6}$

Cancel the common factors.

Factor $2$ out of $6$ .

$y = \frac{x^{2}}{6} + \frac{2 (- 4 x)}{2 (3)} + \frac{\frac{82}{3}}{6}$

Cancel the common factor.

$y = \frac{x^{2}}{6} + \frac{2 (- 4 x)}{2 \cdot 3} + \frac{\frac{82}{3}}{6}$

Rewrite the expression.

$y = \frac{x^{2}}{6} + \frac{- 4 x}{3} + \frac{\frac{82}{3}}{6}$

Move the negative in front of the fraction.

$y = \frac{x^{2}}{6} - \frac{(4) x}{3} + \frac{\frac{82}{3}}{6}$

Multiply the numerator by the reciprocal of the denominator.

$y = \frac{x^{2}}{6} - \frac{4 x}{3} + \frac{82}{3} \cdot \frac{1}{6}$

Cancel the common factor of $2$ .

Factor $2$ out of $82$ .

$y = \frac{x^{2}}{6} - \frac{4 x}{3} + \frac{2 (41)}{3} \cdot \frac{1}{6}$

Factor $2$ out of $6$ .

$y = \frac{x^{2}}{6} - \frac{4 x}{3} + \frac{2 \cdot 41}{3} \cdot \frac{1}{2 \cdot 3}$

Cancel the common factor.

$y = \frac{x^{2}}{6} - \frac{4 x}{3} + \frac{2 \cdot 41}{3} \cdot \frac{1}{2 \cdot 3}$

Rewrite the expression.

$y = \frac{x^{2}}{6} - \frac{4 x}{3} + \frac{41}{3} \cdot \frac{1}{3}$

Multiply $\frac{41}{3}$ by $\frac{1}{3}$ .

$y = \frac{x^{2}}{6} - \frac{4 x}{3} + \frac{41}{3 \cdot 3}$

Multiply $3$ by $3$ .

$y = \frac{x^{2}}{6} - \frac{4 x}{3} + \frac{41}{9}$

Interchange the variables.

$x = \frac{y^{2}}{6} - \frac{4 y}{3} + \frac{41}{9}$

Solve for $y$ .

Rewrite the equation as $\frac{y^{2}}{6} - \frac{4 y}{3} + \frac{41}{9} = x$ .

$\frac{y^{2}}{6} - \frac{4 y}{3} + \frac{41}{9} = x$

Subtract $x$ from both sides of the equation.

$\frac{y^{2}}{6} - \frac{4 y}{3} + \frac{41}{9} - x = 0$

Multiply through by the least common denominator $18$ , then simplify.

Apply the distributive property.

$18 (\frac{y^{2}}{6}) + 18 (- \frac{4 y}{3}) + 18 (\frac{41}{9}) + 18 (- x) = 0$

Simplify.

Cancel the common factor of $6$ .

Factor $6$ out of $18$ .

$6 (3) (\frac{y^{2}}{6}) + 18 (- \frac{4 y}{3}) + 18 (\frac{41}{9}) + 18 (- x) = 0$

Cancel the common factor.

$6 \cdot (3 (\frac{y^{2}}{6})) + 18 (- \frac{4 y}{3}) + 18 (\frac{41}{9}) + 18 (- x) = 0$

Rewrite the expression.

$3 y^{2} + 18 (- \frac{4 y}{3}) + 18 (\frac{41}{9}) + 18 (- x) = 0$

Cancel the common factor of $3$ .

Move the leading negative in $- \frac{4 y}{3}$ into the numerator.

$3 y^{2} + 18 (\frac{- 4 y}{3}) + 18 (\frac{41}{9}) + 18 (- x) = 0$

Factor $3$ out of $18$ .

$3 y^{2} + 3 (6) (\frac{- 4 y}{3}) + 18 (\frac{41}{9}) + 18 (- x) = 0$

Cancel the common factor.

$3 y^{2} + 3 \cdot (6 (\frac{- 4 y}{3})) + 18 (\frac{41}{9}) + 18 (- x) = 0$

Rewrite the expression.

$3 y^{2} + 6 (- 4 y) + 18 (\frac{41}{9}) + 18 (- x) = 0$

Multiply $- 4$ by $6$ .

$3 y^{2} - 24 y + 18 (\frac{41}{9}) + 18 (- x) = 0$

Cancel the common factor of $9$ .

Factor $9$ out of $18$ .

$3 y^{2} - 24 y + 9 (2) (\frac{41}{9}) + 18 (- x) = 0$

Cancel the common factor.

$3 y^{2} - 24 y + 9 \cdot (2 (\frac{41}{9})) + 18 (- x) = 0$

Rewrite the expression.

$3 y^{2} - 24 y + 2 \cdot 41 + 18 (- x) = 0$

Multiply $2$ by $41$ .

$3 y^{2} - 24 y + 82 + 18 (- x) = 0$

Multiply $- 1$ by $18$ .

$3 y^{2} - 24 y + 82 - 18 x = 0$

Move $82$ .

$3 y^{2} - 24 y - 18 x + 82 = 0$

Move $- 24 y$ .

$3 y^{2} - 18 x - 24 y + 82 = 0$

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Substitute the values $a = 3$ , $b = - 24$ , and $c = - 18 x + 82$ into the quadratic formula and solve for $y$ .

$\frac{24 \pm \sqrt{{(- 24)}^{2} - 4 \cdot (3 \cdot (- 18 x + 82))}}{2 \cdot 3}$

Simplify.

Simplify the numerator.

Raise $- 24$ to the power of $2$ .

$y = \frac{24 \pm \sqrt{576 - 4 \cdot 3 \cdot (- 18 x + 82)}}{2 \cdot 3}$

Multiply $- 4$ by $3$ .

$y = \frac{24 \pm \sqrt{576 - 12 \cdot (- 18 x + 82)}}{2 \cdot 3}$

Apply the distributive property.

$y = \frac{24 \pm \sqrt{576 - 12 (- 18 x) - 12 \cdot 82}}{2 \cdot 3}$

Multiply $- 18$ by $- 12$ .

$y = \frac{24 \pm \sqrt{576 + 216 x - 12 \cdot 82}}{2 \cdot 3}$

Multiply $- 12$ by $82$ .

$y = \frac{24 \pm \sqrt{576 + 216 x - 984}}{2 \cdot 3}$

Subtract $984$ from $576$ .

$y = \frac{24 \pm \sqrt{216 x - 408}}{2 \cdot 3}$

Factor $24$ out of $216 x - 408$ .

Factor $24$ out of $216 x$ .

$y = \frac{24 \pm \sqrt{24 (9 x) - 408}}{2 \cdot 3}$

Factor $24$ out of $- 408$ .

$y = \frac{24 \pm \sqrt{24 (9 x) + 24 (- 17)}}{2 \cdot 3}$

Factor $24$ out of $24 (9 x) + 24 (- 17)$ .

$y = \frac{24 \pm \sqrt{24 (9 x - 17)}}{2 \cdot 3}$

Rewrite $24 (9 x - 17)$ as $2^{2} \cdot (6 (3^{2} x - 17))$ .

Factor $4$ out of $24$ .

$y = \frac{24 \pm \sqrt{4 (6) (9 x - 17)}}{2 \cdot 3}$

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

$y = \frac{24 \pm \sqrt{2^{2} \cdot (6 (9 x - 17))}}{2 \cdot 3}$

Rewrite $9$ as $3^{2}$ .

$y = \frac{24 \pm \sqrt{2^{2} \cdot (6 (3^{2} x - 17))}}{2 \cdot 3}$

Add parentheses.

$y = \frac{24 \pm \sqrt{2^{2} \cdot (6 (3^{2} x - 17))}}{2 \cdot 3}$

Pull terms out from under the radical.

$y = \frac{24 \pm 2 \sqrt{6 (3^{2} x - 17)}}{2 \cdot 3}$

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

$y = \frac{24 \pm 2 \sqrt{6 (9 x - 17)}}{2 \cdot 3}$

Multiply $2$ by $3$ .

$y = \frac{24 \pm 2 \sqrt{6 (9 x - 17)}}{6}$

Simplify $\frac{24 \pm 2 \sqrt{6 (9 x - 17)}}{6}$ .

$y = \frac{12 \pm \sqrt{6 (9 x - 17)}}{3}$

Simplify the expression to solve for the $+$ portion of the $\pm$ .

Simplify the numerator.

Raise $- 24$ to the power of $2$ .

$y = \frac{24 \pm \sqrt{576 - 4 \cdot 3 \cdot (- 18 x + 82)}}{2 \cdot 3}$

Multiply $- 4$ by $3$ .

$y = \frac{24 \pm \sqrt{576 - 12 \cdot (- 18 x + 82)}}{2 \cdot 3}$

Apply the distributive property.

$y = \frac{24 \pm \sqrt{576 - 12 (- 18 x) - 12 \cdot 82}}{2 \cdot 3}$

Multiply $- 18$ by $- 12$ .

$y = \frac{24 \pm \sqrt{576 + 216 x - 12 \cdot 82}}{2 \cdot 3}$

Multiply $- 12$ by $82$ .

$y = \frac{24 \pm \sqrt{576 + 216 x - 984}}{2 \cdot 3}$

Subtract $984$ from $576$ .

$y = \frac{24 \pm \sqrt{216 x - 408}}{2 \cdot 3}$

Factor $24$ out of $216 x - 408$ .

Factor $24$ out of $216 x$ .

$y = \frac{24 \pm \sqrt{24 (9 x) - 408}}{2 \cdot 3}$

Factor $24$ out of $- 408$ .

$y = \frac{24 \pm \sqrt{24 (9 x) + 24 (- 17)}}{2 \cdot 3}$

Factor $24$ out of $24 (9 x) + 24 (- 17)$ .

$y = \frac{24 \pm \sqrt{24 (9 x - 17)}}{2 \cdot 3}$

Rewrite $24 (9 x - 17)$ as $2^{2} \cdot (6 (3^{2} x - 17))$ .

Factor $4$ out of $24$ .

$y = \frac{24 \pm \sqrt{4 (6) (9 x - 17)}}{2 \cdot 3}$

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

$y = \frac{24 \pm \sqrt{2^{2} \cdot (6 (9 x - 17))}}{2 \cdot 3}$

Rewrite $9$ as $3^{2}$ .

$y = \frac{24 \pm \sqrt{2^{2} \cdot (6 (3^{2} x - 17))}}{2 \cdot 3}$

Add parentheses.

$y = \frac{24 \pm \sqrt{2^{2} \cdot (6 (3^{2} x - 17))}}{2 \cdot 3}$

Pull terms out from under the radical.

$y = \frac{24 \pm 2 \sqrt{6 (3^{2} x - 17)}}{2 \cdot 3}$

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

$y = \frac{24 \pm 2 \sqrt{6 (9 x - 17)}}{2 \cdot 3}$

Multiply $2$ by $3$ .

$y = \frac{24 \pm 2 \sqrt{6 (9 x - 17)}}{6}$

Simplify $\frac{24 \pm 2 \sqrt{6 (9 x - 17)}}{6}$ .

$y = \frac{12 \pm \sqrt{6 (9 x - 17)}}{3}$

Change the $\pm$ to $+$ .

$y = \frac{12 + \sqrt{6 (9 x - 17)}}{3}$

Simplify the expression to solve for the $-$ portion of the $\pm$ .

Simplify the numerator.

Raise $- 24$ to the power of $2$ .

$y = \frac{24 \pm \sqrt{576 - 4 \cdot 3 \cdot (- 18 x + 82)}}{2 \cdot 3}$

Multiply $- 4$ by $3$ .

$y = \frac{24 \pm \sqrt{576 - 12 \cdot (- 18 x + 82)}}{2 \cdot 3}$

Apply the distributive property.

$y = \frac{24 \pm \sqrt{576 - 12 (- 18 x) - 12 \cdot 82}}{2 \cdot 3}$

Multiply $- 18$ by $- 12$ .

$y = \frac{24 \pm \sqrt{576 + 216 x - 12 \cdot 82}}{2 \cdot 3}$

Multiply $- 12$ by $82$ .

$y = \frac{24 \pm \sqrt{576 + 216 x - 984}}{2 \cdot 3}$

Subtract $984$ from $576$ .

$y = \frac{24 \pm \sqrt{216 x - 408}}{2 \cdot 3}$

Factor $24$ out of $216 x - 408$ .

Factor $24$ out of $216 x$ .

$y = \frac{24 \pm \sqrt{24 (9 x) - 408}}{2 \cdot 3}$

Factor $24$ out of $- 408$ .

$y = \frac{24 \pm \sqrt{24 (9 x) + 24 (- 17)}}{2 \cdot 3}$

Factor $24$ out of $24 (9 x) + 24 (- 17)$ .

$y = \frac{24 \pm \sqrt{24 (9 x - 17)}}{2 \cdot 3}$

Rewrite $24 (9 x - 17)$ as $2^{2} \cdot (6 (3^{2} x - 17))$ .

Factor $4$ out of $24$ .

$y = \frac{24 \pm \sqrt{4 (6) (9 x - 17)}}{2 \cdot 3}$

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

$y = \frac{24 \pm \sqrt{2^{2} \cdot (6 (9 x - 17))}}{2 \cdot 3}$

Rewrite $9$ as $3^{2}$ .

$y = \frac{24 \pm \sqrt{2^{2} \cdot (6 (3^{2} x - 17))}}{2 \cdot 3}$

Add parentheses.

$y = \frac{24 \pm \sqrt{2^{2} \cdot (6 (3^{2} x - 17))}}{2 \cdot 3}$

Pull terms out from under the radical.

$y = \frac{24 \pm 2 \sqrt{6 (3^{2} x - 17)}}{2 \cdot 3}$

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

$y = \frac{24 \pm 2 \sqrt{6 (9 x - 17)}}{2 \cdot 3}$

Multiply $2$ by $3$ .

$y = \frac{24 \pm 2 \sqrt{6 (9 x - 17)}}{6}$

Simplify $\frac{24 \pm 2 \sqrt{6 (9 x - 17)}}{6}$ .

$y = \frac{12 \pm \sqrt{6 (9 x - 17)}}{3}$

Change the $\pm$ to $-$ .

$y = \frac{12 - \sqrt{6 (9 x - 17)}}{3}$

The final answer is the combination of both solutions.

$y = \frac{12 + \sqrt{6 (9 x - 17)}}{3}$

$y = \frac{12 - \sqrt{6 (9 x - 17)}}{3}$

$y = \frac{12 + \sqrt{6 (9 x - 17)}}{3}$

$y = \frac{12 - \sqrt{6 (9 x - 17)}}{3}$

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

$f^{- 1} (x) = \frac{12 + \sqrt{6 (9 x - 17)}}{3}, \frac{12 - \sqrt{6 (9 x - 17)}}{3}$

Verify if $f^{- 1} (x) = \frac{12 + \sqrt{6 (9 x - 17)}}{3}, \frac{12 - \sqrt{6 (9 x - 17)}}{3}$ is the inverse of $f (x) = \frac{x^{2}}{6} - \frac{4 x}{3} + \frac{41}{9}$ .

The domain of the inverse is the range of the original function and vice versa. Find the domain and the range of $f (x) = \frac{x^{2}}{6} - \frac{4 x}{3} + \frac{41}{9}$ and $f^{- 1} (x) = \frac{12 + \sqrt{6 (9 x - 17)}}{3}, \frac{12 - \sqrt{6 (9 x - 17)}}{3}$ and compare them.

Find the range of $f (x) = \frac{x^{2}}{6} - \frac{4 x}{3} + \frac{41}{9}$ .

The range is the set of all valid $y$ values. Use the graph to find the range.

Interval Notation:

$[\frac{17}{9}, \infty)$

Find the domain of $\frac{12 + \sqrt{6 (9 x - 17)}}{3}$ .

Set the radicand in $\sqrt{6 (9 x - 17)}$ greater than or equal to $0$ to find where the expression is defined.

$6 (9 x - 17) \geq 0$

Solve for $x$ .

Divide each term in $6 (9 x - 17) \geq 0$ by $6$ and simplify.

Divide each term in $6 (9 x - 17) \geq 0$ by $6$ .

$\frac{6 (9 x - 17)}{6} \geq \frac{0}{6}$

Simplify the left side.

Cancel the common factor of $6$ .

Cancel the common factor.

$\frac{6 (9 x - 17)}{6} \geq \frac{0}{6}$

Divide $9 x - 17$ by $1$ .

$9 x - 17 \geq \frac{0}{6}$

Simplify the right side.

Divide $0$ by $6$ .

$9 x - 17 \geq 0$

Add $17$ to both sides of the inequality.

$9 x \geq 17$

Divide each term in $9 x \geq 17$ by $9$ and simplify.

Divide each term in $9 x \geq 17$ by $9$ .

$\frac{9 x}{9} \geq \frac{17}{9}$

Simplify the left side.

Cancel the common factor of $9$ .

Cancel the common factor.

$\frac{9 x}{9} \geq \frac{17}{9}$

Divide $x$ by $1$ .

$x \geq \frac{17}{9}$

The domain is all values of $x$ that make the expression defined.

$[\frac{17}{9}, \infty)$

Find the domain of $f (x) = \frac{x^{2}}{6} - \frac{4 x}{3} + \frac{41}{9}$ .

The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.

$(- \infty, \infty)$

Since the domain of $f^{- 1} (x) = \frac{12 + \sqrt{6 (9 x - 17)}}{3}, \frac{12 - \sqrt{6 (9 x - 17)}}{3}$ is the range of $f (x) = \frac{x^{2}}{6} - \frac{4 x}{3} + \frac{41}{9}$ and the range of $f^{- 1} (x) = \frac{12 + \sqrt{6 (9 x - 17)}}{3}, \frac{12 - \sqrt{6 (9 x - 17)}}{3}$ is the domain of $f (x) = \frac{x^{2}}{6} - \frac{4 x}{3} + \frac{41}{9}$ , then $f^{- 1} (x) = \frac{12 + \sqrt{6 (9 x - 17)}}{3}, \frac{12 - \sqrt{6 (9 x - 17)}}{3}$ is the inverse of $f (x) = \frac{x^{2}}{6} - \frac{4 x}{3} + \frac{41}{9}$ .

$f^{- 1} (x) = \frac{12 + \sqrt{6 (9 x - 17)}}{3}, \frac{12 - \sqrt{6 (9 x - 17)}}{3}$

Find the Inverse (x-4)^2-2/3=6y-12

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