Algebra Examples

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

Step 1

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

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

Step 2

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

Tap for more steps...

Step 2.1

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

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

Step 2.2

Simplify the left side.

Tap for more steps...

Step 2.2.1

Cancel the common factor of $8$ .

Tap for more steps...

Step 2.2.1.1

Cancel the common factor.

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

Step 2.2.1.2

Divide $7 + y$ by $1$ .

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

Step 2.3

Simplify the right side.

Tap for more steps...

Step 2.3.1

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

Tap for more steps...

Step 2.3.1.1

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

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

Step 2.3.1.2

Cancel the common factors.

Tap for more steps...

Step 2.3.1.2.1

Factor $2$ out of $8$ .

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

Step 2.3.1.2.2

Cancel the common factor.

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

Step 2.3.1.2.3

Rewrite the expression.

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

Step 3

Subtract $7$ from both sides of the equation.

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

Step 4

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

Tap for more steps...

Step 4.1

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

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

Step 4.2

Simplify terms.

Tap for more steps...

Step 4.2.1

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

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

Step 4.2.2

Combine the numerators over the common denominator.

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

Step 4.3

Simplify the numerator.

Tap for more steps...

Step 4.3.1

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

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

Step 4.3.2

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

Tap for more steps...

Step 4.3.2.1

Apply the distributive property.

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

Step 4.3.2.2

Apply the distributive property.

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

Step 4.3.2.3

Apply the distributive property.

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

Step 4.3.3

Simplify and combine like terms.

Tap for more steps...

Step 4.3.3.1

Simplify each term.

Tap for more steps...

Step 4.3.3.1.1

Multiply $x$ by $x$ .

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

Step 4.3.3.1.2

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

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

Step 4.3.3.1.3

Multiply $- 2$ by $- 2$ .

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

Step 4.3.3.2

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

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

Step 4.3.4

Multiply $- 7$ by $4$ .

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

Step 4.3.5

Subtract $28$ from $4$ .

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

Step 5

Interchange the variables.

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

Step 6

Solve for $y$ .

Tap for more steps...

Step 6.1

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

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

Step 6.2

Multiply both sides by $4$ .

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

Step 6.3

Simplify.

Tap for more steps...

Step 6.3.1

Simplify the left side.

Tap for more steps...

Step 6.3.1.1

Cancel the common factor of $4$ .

Tap for more steps...

Step 6.3.1.1.1

Cancel the common factor.

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

Step 6.3.1.1.2

Rewrite the expression.

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

Step 6.3.2

Simplify the right side.

Tap for more steps...

Step 6.3.2.1

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

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

Step 6.4

Solve for $y$ .

Tap for more steps...

Step 6.4.1

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

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

Step 6.4.2

Use the quadratic formula to find the solutions.

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

Step 6.4.3

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

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

Step 6.4.4

Simplify.

Tap for more steps...

Step 6.4.4.1

Simplify the numerator.

Tap for more steps...

Step 6.4.4.1.1

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

Tap for more steps...

Step 6.4.4.1.1.1

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

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

Step 6.4.4.1.1.2

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

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

Step 6.4.4.1.1.3

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

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

Step 6.4.4.1.2

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

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

Step 6.4.4.1.3

Subtract $24$ from $- 4$ .

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

Step 6.4.4.1.4

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

Tap for more steps...

Step 6.4.4.1.4.1

Factor $4$ out of $- 28$ .

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

Step 6.4.4.1.4.2

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

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

Step 6.4.4.1.4.3

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

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

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

Step 6.4.4.1.5

Multiply $- 4$ by $4$ .

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

Step 6.4.4.1.6

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

Tap for more steps...

Step 6.4.4.1.6.1

Factor $16$ out of $- 16$ .

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

Step 6.4.4.1.6.2

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

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

Step 6.4.4.1.6.3

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

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

Step 6.4.4.1.6.4

Add parentheses.

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

Step 6.4.4.1.7

Pull terms out from under the radical.

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

Step 6.4.4.1.8

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

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

Step 6.4.4.2

Multiply $2$ by $1$ .

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

Step 6.4.4.3

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

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

Step 6.4.5

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

Tap for more steps...

Step 6.4.5.1

Simplify the numerator.

Tap for more steps...

Step 6.4.5.1.1

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

Tap for more steps...

Step 6.4.5.1.1.1

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

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

Step 6.4.5.1.1.2

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

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

Step 6.4.5.1.1.3

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

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

Step 6.4.5.1.2

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

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

Step 6.4.5.1.3

Subtract $24$ from $- 4$ .

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

Step 6.4.5.1.4

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

Tap for more steps...

Step 6.4.5.1.4.1

Factor $4$ out of $- 28$ .

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

Step 6.4.5.1.4.2

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

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

Step 6.4.5.1.4.3

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

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

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

Step 6.4.5.1.5

Multiply $- 4$ by $4$ .

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

Step 6.4.5.1.6

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

Tap for more steps...

Step 6.4.5.1.6.1

Factor $16$ out of $- 16$ .

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

Step 6.4.5.1.6.2

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

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

Step 6.4.5.1.6.3

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

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

Step 6.4.5.1.6.4

Add parentheses.

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

Step 6.4.5.1.7

Pull terms out from under the radical.

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

Step 6.4.5.1.8

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

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

Step 6.4.5.2

Multiply $2$ by $1$ .

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

Step 6.4.5.3

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

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

Step 6.4.5.4

Change the $\pm$ to $+$ .

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

Step 6.4.6

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

Tap for more steps...

Step 6.4.6.1

Simplify the numerator.

Tap for more steps...

Step 6.4.6.1.1

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

Tap for more steps...

Step 6.4.6.1.1.1

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

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

Step 6.4.6.1.1.2

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

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

Step 6.4.6.1.1.3

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

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

Step 6.4.6.1.2

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

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

Step 6.4.6.1.3

Subtract $24$ from $- 4$ .

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

Step 6.4.6.1.4

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

Tap for more steps...

Step 6.4.6.1.4.1

Factor $4$ out of $- 28$ .

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

Step 6.4.6.1.4.2

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

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

Step 6.4.6.1.4.3

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

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

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

Step 6.4.6.1.5

Multiply $- 4$ by $4$ .

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

Step 6.4.6.1.6

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

Tap for more steps...

Step 6.4.6.1.6.1

Factor $16$ out of $- 16$ .

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

Step 6.4.6.1.6.2

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

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

Step 6.4.6.1.6.3

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

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

Step 6.4.6.1.6.4

Add parentheses.

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

Step 6.4.6.1.7

Pull terms out from under the radical.

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

Step 6.4.6.1.8

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

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

Step 6.4.6.2

Multiply $2$ by $1$ .

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

Step 6.4.6.3

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

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

Step 6.4.6.4

Change the $\pm$ to $-$ .

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

Step 6.4.7

The final answer is the combination of both solutions.

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

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

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

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

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

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

Step 7

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

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

Step 8

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

Tap for more steps...

Step 8.1

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.

Step 8.2

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

Tap for more steps...

Step 8.2.1

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

Interval Notation:

$[- 7, \infty)$

Step 8.3

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

Tap for more steps...

Step 8.3.1

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

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

Step 8.3.2

Solve for $x$ .

Tap for more steps...

Step 8.3.2.1

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

Tap for more steps...

Step 8.3.2.1.1

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.

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

Step 8.3.2.1.2

Simplify the left side.

Tap for more steps...

Step 8.3.2.1.2.1

Dividing two negative values results in a positive value.

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

Step 8.3.2.1.2.2

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

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

Step 8.3.2.1.3

Simplify the right side.

Tap for more steps...

Step 8.3.2.1.3.1

Divide $0$ by $- 1$ .

$- 7 - x \leq 0$

Step 8.3.2.2

Add $7$ to both sides of the inequality.

$- x \leq 7$

Step 8.3.2.3

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

Tap for more steps...

Step 8.3.2.3.1

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.

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

Step 8.3.2.3.2

Simplify the left side.

Tap for more steps...

Step 8.3.2.3.2.1

Dividing two negative values results in a positive value.

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

Step 8.3.2.3.2.2

Divide $x$ by $1$ .

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

Step 8.3.2.3.3

Simplify the right side.

Tap for more steps...

Step 8.3.2.3.3.1

Divide $7$ by $- 1$ .

$x \geq - 7$

Step 8.3.3

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

$[- 7, \infty)$

Step 8.4

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

Tap for more steps...

Step 8.4.1

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.

$(- \infty, \infty)$

Step 8.5

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

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

Step 9