Find the Inverse 2(x-2)^2=8(7+y)

$2 {(x - 2)}^{2} = 8 (7 + y)$

Rewrite the equation as $8 (7 + y) = 2 {(x - 2)}^{2}$ .

$8 (7 + y) = 2 {(x - 2)}^{2}$

Divide each term in $8 (7 + y) = 2 {(x - 2)}^{2}$ by $8$ and simplify.

Divide each term in $8 (7 + y) = 2 {(x - 2)}^{2}$ by $8$ .

$\frac{8 (7 + y)}{8} = \frac{2 {(x - 2)}^{2}}{8}$

Simplify the left side.

Cancel the common factor of $8$ .

Cancel the common factor.

$\frac{8 (7 + y)}{8} = \frac{2 {(x - 2)}^{2}}{8}$

Divide $7 + y$ by $1$ .

$7 + y = \frac{2 {(x - 2)}^{2}}{8}$

Simplify the right side.

Cancel the common factor of $2$ and $8$ .

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

$7 + y = \frac{2 ({(x - 2)}^{2})}{8}$

Cancel the common factors.

Factor $2$ out of $8$ .

$7 + y = \frac{2 {(x - 2)}^{2}}{2 \cdot 4}$

Cancel the common factor.

$7 + y = \frac{2 {(x - 2)}^{2}}{2 \cdot 4}$

Rewrite the expression.

$7 + y = \frac{{(x - 2)}^{2}}{4}$

Subtract $7$ from both sides of the equation.

$y = \frac{{(x - 2)}^{2}}{4} - 7$

Simplify $\frac{{(x - 2)}^{2}}{4} - 7$ .

To write $- 7$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$y = \frac{{(x - 2)}^{2}}{4} - 7 \cdot \frac{4}{4}$

Simplify terms.

Combine $- 7$ and $\frac{4}{4}$ .

$y = \frac{{(x - 2)}^{2}}{4} + \frac{- 7 \cdot 4}{4}$

Combine the numerators over the common denominator.

$y = \frac{{(x - 2)}^{2} - 7 \cdot 4}{4}$

Simplify the numerator.

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

$y = \frac{(x - 2) (x - 2) - 7 \cdot 4}{4}$

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

Apply the distributive property.

$y = \frac{x (x - 2) - 2 (x - 2) - 7 \cdot 4}{4}$

Apply the distributive property.

$y = \frac{x \cdot x + x \cdot - 2 - 2 (x - 2) - 7 \cdot 4}{4}$

Apply the distributive property.

$y = \frac{x \cdot x + x \cdot - 2 - 2 x - 2 \cdot - 2 - 7 \cdot 4}{4}$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$y = \frac{x^{2} + x \cdot - 2 - 2 x - 2 \cdot - 2 - 7 \cdot 4}{4}$

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

$y = \frac{x^{2} - 2 \cdot x - 2 x - 2 \cdot - 2 - 7 \cdot 4}{4}$

Multiply $- 2$ by $- 2$ .

$y = \frac{x^{2} - 2 x - 2 x + 4 - 7 \cdot 4}{4}$

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

$y = \frac{x^{2} - 4 x + 4 - 7 \cdot 4}{4}$

Multiply $- 7$ by $4$ .

$y = \frac{x^{2} - 4 x + 4 - 28}{4}$

Subtract $28$ from $4$ .

$y = \frac{x^{2} - 4 x - 24}{4}$

Interchange the variables.

$x = \frac{y^{2} - 4 y - 24}{4}$

Solve for $y$ .

Rewrite the equation as $\frac{y^{2} - 4 y - 24}{4} = x$ .

$\frac{y^{2} - 4 y - 24}{4} = x$

Multiply both sides by $4$ .

$\frac{y^{2} - 4 y - 24}{4} \cdot 4 = x \cdot 4$

Simplify.

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{y^{2} - 4 y - 24}{4} \cdot 4 = x \cdot 4$

Rewrite the expression.

$y^{2} - 4 y - 24 = x \cdot 4$

Simplify the right side.

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

$y^{2} - 4 y - 24 = 4 x$

Solve for $y$ .

Subtract $4 x$ from both sides of the equation.

$y^{2} - 4 y - 24 - 4 x = 0$

Use the quadratic formula to find the solutions.

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

Substitute the values $a = 1$ , $b = - 4$ , and $c = - 24 - 4 x$ into the quadratic formula and solve for $y$ .

$\frac{4 \pm \sqrt{{(- 4)}^{2} - 4 \cdot (1 \cdot (- 24 - 4 x))}}{2 \cdot 1}$

Simplify.

Simplify the numerator.

Factor $- 4$ out of ${(- 4)}^{2} - 4 \cdot 1 \cdot (- 24 - 4 x)$ .

Factor $- 4$ out of ${(- 4)}^{2}$ .

$y = \frac{4 \pm \sqrt{- 4 \cdot - 4 - 4 \cdot 1 \cdot (- 24 - 4 x)}}{2 \cdot 1}$

Factor $- 4$ out of $- 4 \cdot 1 \cdot (- 24 - 4 x)$ .

$y = \frac{4 \pm \sqrt{- 4 \cdot - 4 - 4 (1 \cdot (- 24 - 4 x))}}{2 \cdot 1}$

Factor $- 4$ out of $- 4 \cdot - 4 - 4 (1 \cdot (- 24 - 4 x))$ .

$y = \frac{4 \pm \sqrt{- 4 (- 4 + 1 \cdot (- 24 - 4 x))}}{2 \cdot 1}$

Multiply $- 24 - 4 x$ by $1$ .

$y = \frac{4 \pm \sqrt{- 4 (- 4 - 24 - 4 x)}}{2 \cdot 1}$

Subtract $24$ from $- 4$ .

$y = \frac{4 \pm \sqrt{- 4 (- 28 - 4 x)}}{2 \cdot 1}$

Factor $4$ out of $- 28 - 4 x$ .

Factor $4$ out of $- 28$ .

$y = \frac{4 \pm \sqrt{- 4 (4 (- 7) - 4 x)}}{2 \cdot 1}$

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

$y = \frac{4 \pm \sqrt{- 4 (4 (- 7) + 4 (- x))}}{2 \cdot 1}$

Factor $4$ out of $4 (- 7) + 4 (- x)$ .

$y = \frac{4 \pm \sqrt{- 4 (4 (- 7 - x))}}{2 \cdot 1}$

$y = \frac{4 \pm \sqrt{- 4 \cdot (4 (- 7 - x))}}{2 \cdot 1}$

Multiply $- 4$ by $4$ .

$y = \frac{4 \pm \sqrt{- 16 (- 7 - x)}}{2 \cdot 1}$

Rewrite $- 16 (- 7 - x)$ as ${(2^{2})}^{2} \cdot (- 1 (- 7 - x))$ .

Factor $16$ out of $- 16$ .

$y = \frac{4 \pm \sqrt{16 (- 1) (- 7 - x)}}{2 \cdot 1}$

Rewrite $16$ as $4^{2}$ .

$y = \frac{4 \pm \sqrt{4^{2} \cdot (- 1 (- 7 - x))}}{2 \cdot 1}$

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

$y = \frac{4 \pm \sqrt{{(2^{2})}^{2} \cdot (- 1 (- 7 - x))}}{2 \cdot 1}$

Add parentheses.

$y = \frac{4 \pm \sqrt{{(2^{2})}^{2} \cdot (- 1 (- 7 - x))}}{2 \cdot 1}$

Pull terms out from under the radical.

$y = \frac{4 \pm 2^{2} \sqrt{- 1 (- 7 - x)}}{2 \cdot 1}$

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

$y = \frac{4 \pm 4 \sqrt{- 1 (- 7 - x)}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$y = \frac{4 \pm 4 \sqrt{- 1 (- 7 - x)}}{2}$

Simplify $\frac{4 \pm 4 \sqrt{- 1 (- 7 - x)}}{2}$ .

$y = 2 \pm 2 \sqrt{- 1 (- 7 - x)}$

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

Simplify the numerator.

Factor $- 4$ out of ${(- 4)}^{2} - 4 \cdot 1 \cdot (- 24 - 4 x)$ .

Factor $- 4$ out of ${(- 4)}^{2}$ .

$y = \frac{4 \pm \sqrt{- 4 \cdot - 4 - 4 \cdot 1 \cdot (- 24 - 4 x)}}{2 \cdot 1}$

Factor $- 4$ out of $- 4 \cdot 1 \cdot (- 24 - 4 x)$ .

$y = \frac{4 \pm \sqrt{- 4 \cdot - 4 - 4 (1 \cdot (- 24 - 4 x))}}{2 \cdot 1}$

Factor $- 4$ out of $- 4 \cdot - 4 - 4 (1 \cdot (- 24 - 4 x))$ .

$y = \frac{4 \pm \sqrt{- 4 (- 4 + 1 \cdot (- 24 - 4 x))}}{2 \cdot 1}$

Multiply $- 24 - 4 x$ by $1$ .

$y = \frac{4 \pm \sqrt{- 4 (- 4 - 24 - 4 x)}}{2 \cdot 1}$

Subtract $24$ from $- 4$ .

$y = \frac{4 \pm \sqrt{- 4 (- 28 - 4 x)}}{2 \cdot 1}$

Factor $4$ out of $- 28 - 4 x$ .

Factor $4$ out of $- 28$ .

$y = \frac{4 \pm \sqrt{- 4 (4 (- 7) - 4 x)}}{2 \cdot 1}$

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

$y = \frac{4 \pm \sqrt{- 4 (4 (- 7) + 4 (- x))}}{2 \cdot 1}$

Factor $4$ out of $4 (- 7) + 4 (- x)$ .

$y = \frac{4 \pm \sqrt{- 4 (4 (- 7 - x))}}{2 \cdot 1}$

$y = \frac{4 \pm \sqrt{- 4 \cdot (4 (- 7 - x))}}{2 \cdot 1}$

Multiply $- 4$ by $4$ .

$y = \frac{4 \pm \sqrt{- 16 (- 7 - x)}}{2 \cdot 1}$

Rewrite $- 16 (- 7 - x)$ as ${(2^{2})}^{2} \cdot (- 1 (- 7 - x))$ .

Factor $16$ out of $- 16$ .

$y = \frac{4 \pm \sqrt{16 (- 1) (- 7 - x)}}{2 \cdot 1}$

Rewrite $16$ as $4^{2}$ .

$y = \frac{4 \pm \sqrt{4^{2} \cdot (- 1 (- 7 - x))}}{2 \cdot 1}$

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

$y = \frac{4 \pm \sqrt{{(2^{2})}^{2} \cdot (- 1 (- 7 - x))}}{2 \cdot 1}$

Add parentheses.

$y = \frac{4 \pm \sqrt{{(2^{2})}^{2} \cdot (- 1 (- 7 - x))}}{2 \cdot 1}$

Pull terms out from under the radical.

$y = \frac{4 \pm 2^{2} \sqrt{- 1 (- 7 - x)}}{2 \cdot 1}$

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

$y = \frac{4 \pm 4 \sqrt{- 1 (- 7 - x)}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$y = \frac{4 \pm 4 \sqrt{- 1 (- 7 - x)}}{2}$

Simplify $\frac{4 \pm 4 \sqrt{- 1 (- 7 - x)}}{2}$ .

$y = 2 \pm 2 \sqrt{- 1 (- 7 - x)}$

Change the $\pm$ to $+$ .

$y = 2 + 2 \sqrt{- 1 (- 7 - x)}$

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

Simplify the numerator.

Factor $- 4$ out of ${(- 4)}^{2} - 4 \cdot 1 \cdot (- 24 - 4 x)$ .

Factor $- 4$ out of ${(- 4)}^{2}$ .

$y = \frac{4 \pm \sqrt{- 4 \cdot - 4 - 4 \cdot 1 \cdot (- 24 - 4 x)}}{2 \cdot 1}$

Factor $- 4$ out of $- 4 \cdot 1 \cdot (- 24 - 4 x)$ .

$y = \frac{4 \pm \sqrt{- 4 \cdot - 4 - 4 (1 \cdot (- 24 - 4 x))}}{2 \cdot 1}$

Factor $- 4$ out of $- 4 \cdot - 4 - 4 (1 \cdot (- 24 - 4 x))$ .

$y = \frac{4 \pm \sqrt{- 4 (- 4 + 1 \cdot (- 24 - 4 x))}}{2 \cdot 1}$

Multiply $- 24 - 4 x$ by $1$ .

$y = \frac{4 \pm \sqrt{- 4 (- 4 - 24 - 4 x)}}{2 \cdot 1}$

Subtract $24$ from $- 4$ .

$y = \frac{4 \pm \sqrt{- 4 (- 28 - 4 x)}}{2 \cdot 1}$

Factor $4$ out of $- 28 - 4 x$ .

Factor $4$ out of $- 28$ .

$y = \frac{4 \pm \sqrt{- 4 (4 (- 7) - 4 x)}}{2 \cdot 1}$

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

$y = \frac{4 \pm \sqrt{- 4 (4 (- 7) + 4 (- x))}}{2 \cdot 1}$

Factor $4$ out of $4 (- 7) + 4 (- x)$ .

$y = \frac{4 \pm \sqrt{- 4 (4 (- 7 - x))}}{2 \cdot 1}$

$y = \frac{4 \pm \sqrt{- 4 \cdot (4 (- 7 - x))}}{2 \cdot 1}$

Multiply $- 4$ by $4$ .

$y = \frac{4 \pm \sqrt{- 16 (- 7 - x)}}{2 \cdot 1}$

Rewrite $- 16 (- 7 - x)$ as ${(2^{2})}^{2} \cdot (- 1 (- 7 - x))$ .

Factor $16$ out of $- 16$ .

$y = \frac{4 \pm \sqrt{16 (- 1) (- 7 - x)}}{2 \cdot 1}$

Rewrite $16$ as $4^{2}$ .

$y = \frac{4 \pm \sqrt{4^{2} \cdot (- 1 (- 7 - x))}}{2 \cdot 1}$

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

$y = \frac{4 \pm \sqrt{{(2^{2})}^{2} \cdot (- 1 (- 7 - x))}}{2 \cdot 1}$

Add parentheses.

$y = \frac{4 \pm \sqrt{{(2^{2})}^{2} \cdot (- 1 (- 7 - x))}}{2 \cdot 1}$

Pull terms out from under the radical.

$y = \frac{4 \pm 2^{2} \sqrt{- 1 (- 7 - x)}}{2 \cdot 1}$

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

$y = \frac{4 \pm 4 \sqrt{- 1 (- 7 - x)}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$y = \frac{4 \pm 4 \sqrt{- 1 (- 7 - x)}}{2}$

Simplify $\frac{4 \pm 4 \sqrt{- 1 (- 7 - x)}}{2}$ .

$y = 2 \pm 2 \sqrt{- 1 (- 7 - x)}$

Change the $\pm$ to $-$ .

$y = 2 - 2 \sqrt{- 1 (- 7 - x)}$

The final answer is the combination of both solutions.

$y = 2 + 2 \sqrt{- 1 (- 7 - x)}$

$y = 2 - 2 \sqrt{- 1 (- 7 - x)}$

$y = 2 + 2 \sqrt{- 1 (- 7 - x)}$

$y = 2 - 2 \sqrt{- 1 (- 7 - x)}$

$y = 2 + 2 \sqrt{- 1 (- 7 - x)}$

$y = 2 - 2 \sqrt{- 1 (- 7 - x)}$

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

$f^{- 1} (x) = 2 + 2 \sqrt{- 1 (- 7 - x)}, 2 - 2 \sqrt{- 1 (- 7 - x)}$

Verify if $f^{- 1} (x) = 2 + 2 \sqrt{- 1 (- 7 - x)}, 2 - 2 \sqrt{- 1 (- 7 - x)}$ is the inverse of $f (x) = \frac{x^{2} - 4 x - 24}{4}$ .

The domain of the inverse is the range of the original function and vice versa. Find the domain and the range of $f (x) = \frac{x^{2} - 4 x - 24}{4}$ and $f^{- 1} (x) = 2 + 2 \sqrt{- 1 (- 7 - x)}, 2 - 2 \sqrt{- 1 (- 7 - x)}$ and compare them.

Find the range of $f (x) = \frac{x^{2} - 4 x - 24}{4}$ .

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

Interval Notation:

$[- 7, \infty)$

Find the domain of $2 + 2 \sqrt{- 1 (- 7 - x)}$ .

Set the radicand in $\sqrt{- 1 (- 7 - x)}$ greater than or equal to $0$ to find where the expression is defined.

$- 1 (- 7 - x) \geq 0$

Solve for $x$ .

Divide each term in $- 1 (- 7 - x) \geq 0$ by $- 1$ and simplify.

Divide each term in $- 1 (- 7 - x) \geq 0$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 1 (- 7 - x)}{- 1} \leq \frac{0}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{- 7 - x}{1} \leq \frac{0}{- 1}$

Divide $- 7 - x$ by $1$ .

$- 7 - x \leq \frac{0}{- 1}$

Simplify the right side.

Divide $0$ by $- 1$ .

$- 7 - x \leq 0$

Add $7$ to both sides of the inequality.

$- x \leq 7$

Divide each term in $- x \leq 7$ by $- 1$ and simplify.

Divide each term in $- x \leq 7$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x}{- 1} \geq \frac{7}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{x}{1} \geq \frac{7}{- 1}$

Divide $x$ by $1$ .

$x \geq \frac{7}{- 1}$

Simplify the right side.

Divide $7$ by $- 1$ .

$x \geq - 7$

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

$[- 7, \infty)$

Find the domain of $f (x) = \frac{x^{2} - 4 x - 24}{4}$ .

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 $f^{- 1} (x) = 2 + 2 \sqrt{- 1 (- 7 - x)}, 2 - 2 \sqrt{- 1 (- 7 - x)}$ is the range of $f (x) = \frac{x^{2} - 4 x - 24}{4}$ and the range of $f^{- 1} (x) = 2 + 2 \sqrt{- 1 (- 7 - x)}, 2 - 2 \sqrt{- 1 (- 7 - x)}$ is the domain of $f (x) = \frac{x^{2} - 4 x - 24}{4}$ , then $f^{- 1} (x) = 2 + 2 \sqrt{- 1 (- 7 - x)}, 2 - 2 \sqrt{- 1 (- 7 - x)}$ is the inverse of $f (x) = \frac{x^{2} - 4 x - 24}{4}$ .

$f^{- 1} (x) = 2 + 2 \sqrt{- 1 (- 7 - x)}, 2 - 2 \sqrt{- 1 (- 7 - x)}$

Find the Inverse 2(x-2)^2=8(7+y)

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