Find the x and y Intercepts f(x) = square root of -x-2-2

$f (x) = \sqrt{- x - 2} - 2$

Step 1

Find the x-intercepts.

To find the x-intercept(s), substitute in $0$ for $y$ and solve for $x$ .

$0 = \sqrt{- x - 2} - 2$

Solve the equation.

Rewrite the equation as $\sqrt{- x - 2} - 2 = 0$ .

$\sqrt{- x - 2} - 2 = 0$

Add $2$ to both sides of the equation.

$\sqrt{- x - 2} = 2$

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

${\sqrt{- x - 2}}^{2} = 2^{2}$

Simplify each side of the equation.

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

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

Simplify the left side.

Simplify ${({(- x - 2)}^{\frac{1}{2}})}^{2}$ .

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

${(- x - 2)}^{1} = 2^{2}$

Simplify.

$- x - 2 = 2^{2}$

Simplify the right side.

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

$- x - 2 = 4$

Solve for $x$ .

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

Add $2$ to both sides of the equation.

$- x = 4 + 2$

Add $4$ and $2$ .

$- x = 6$

Divide each term in $- x = 6$ by $- 1$ and simplify.

Divide each term in $- x = 6$ by $- 1$ .

$\frac{- x}{- 1} = \frac{6}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{6}{- 1}$

Divide $x$ by $1$ .

$x = \frac{6}{- 1}$

Simplify the right side.

Divide $6$ by $- 1$ .

$x = - 6$

x-intercept(s) in point form.

x-intercept(s): $(- 6, 0)$

Step 2

Find the y-intercepts.

To find the y-intercept(s), substitute in $0$ for $x$ and solve for $y$ .

$y = \sqrt{- (0) - 2} - 2$

Simplify $\sqrt{- (0) - 2} - 2$ .

Simplify each term.

Multiply $- 1$ by $0$ .

$y = \sqrt{0 - 2} - 2$

Subtract $2$ from $0$ .

$y = \sqrt{- 2} - 2$

Rewrite $- 2$ as $- 1 (2)$ .

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

Rewrite $\sqrt{- 1 (2)}$ as $\sqrt{- 1} \cdot \sqrt{2}$ .

$y = \sqrt{- 1} \cdot \sqrt{2} - 2$

Rewrite $\sqrt{- 1}$ as $i$ .

$y = i \sqrt{2} - 2$

Reorder $i \sqrt{2}$ and $- 2$ .

$y = - 2 + i \sqrt{2}$

To find the y-intercept(s), substitute in $0$ for $x$ and solve for $y$ .

y-intercept(s): $None$

Step 3

List the intersections.

x-intercept(s): $(- 6, 0)$

y-intercept(s): $None$

Step 4

Find the x and y Intercepts f(x) = square root of -x-2-2

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