Find the Inverse f(x)=-3 square root of x+2-6 , x>=-2

$f (x) = - 3 \sqrt{x + 2} - 6$ , $x \geq - 2$

Find the range of the given function.

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

$(- \infty, - 6]$

Convert $(- \infty, - 6]$ to an inequality.

$y \leq - 6$

Find the inverse.

Interchange the variables.

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

Solve for $y$ .

Rewrite the equation as $- 3 \sqrt{y + 2} - 6 = x$ .

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

Add $6$ to both sides of the equation.

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

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

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

Simplify each side of the equation.

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

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

Simplify the left side.

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

Apply the product rule to $- 3 {(y + 2)}^{\frac{1}{2}}$ .

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

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

$9 {({(y + 2)}^{\frac{1}{2}})}^{2} = {(x + 6)}^{2}$

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

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

$9 {(y + 2)}^{\frac{1}{2} \cdot 2} = {(x + 6)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$9 {(y + 2)}^{\frac{1}{2} \cdot 2} = {(x + 6)}^{2}$

Rewrite the expression.

$9 {(y + 2)}^{1} = {(x + 6)}^{2}$

Simplify.

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

Apply the distributive property.

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

Multiply $9$ by $2$ .

$9 y + 18 = {(x + 6)}^{2}$

Simplify the right side.

Simplify ${(x + 6)}^{2}$ .

Rewrite ${(x + 6)}^{2}$ as $(x + 6) (x + 6)$ .

$9 y + 18 = (x + 6) (x + 6)$

Expand $(x + 6) (x + 6)$ using the FOIL Method.

Apply the distributive property.

$9 y + 18 = x (x + 6) + 6 (x + 6)$

Apply the distributive property.

$9 y + 18 = x \cdot x + x \cdot 6 + 6 (x + 6)$

Apply the distributive property.

$9 y + 18 = x \cdot x + x \cdot 6 + 6 x + 6 \cdot 6$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$9 y + 18 = x^{2} + x \cdot 6 + 6 x + 6 \cdot 6$

Move $6$ to the left of $x$ .

$9 y + 18 = x^{2} + 6 \cdot x + 6 x + 6 \cdot 6$

Multiply $6$ by $6$ .

$9 y + 18 = x^{2} + 6 x + 6 x + 36$

Add $6 x$ and $6 x$ .

$9 y + 18 = x^{2} + 12 x + 36$

Solve for $y$ .

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

Subtract $18$ from both sides of the equation.

$9 y = x^{2} + 12 x + 36 - 18$

Subtract $18$ from $36$ .

$9 y = x^{2} + 12 x + 18$

Divide each term in $9 y = x^{2} + 12 x + 18$ by $9$ and simplify.

Divide each term in $9 y = x^{2} + 12 x + 18$ by $9$ .

$\frac{9 y}{9} = \frac{x^{2}}{9} + \frac{12 x}{9} + \frac{18}{9}$

Simplify the left side.

Cancel the common factor of $9$ .

Cancel the common factor.

$\frac{9 y}{9} = \frac{x^{2}}{9} + \frac{12 x}{9} + \frac{18}{9}$

Divide $y$ by $1$ .

$y = \frac{x^{2}}{9} + \frac{12 x}{9} + \frac{18}{9}$

Simplify the right side.

Simplify each term.

Cancel the common factor of $12$ and $9$ .

Factor $3$ out of $12 x$ .

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

Cancel the common factors.

Factor $3$ out of $9$ .

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

Cancel the common factor.

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

Rewrite the expression.

$y = \frac{x^{2}}{9} + \frac{4 x}{3} + \frac{18}{9}$

Divide $18$ by $9$ .

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

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

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

Find the inverse using the domain and the range of the original function.

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

Find the Inverse f(x)=-3 square root of x+2-6 , x>=-2

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