Pre-Algebra Examples

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

Step 1

Find the domain for $y = - \sqrt{- (x + 2)} - 3$ so that a list of $x$ values can be picked to find a list of points, which will help graphing the radical.

Tap for more steps...

Step 1.1

Set the radicand in $\sqrt{- (x + 2)}$ greater than or equal to $0$ to find where the expression is defined.

$- (x + 2) \geq 0$

Step 1.2

Solve for $x$ .

Tap for more steps...

Step 1.2.1

Simplify $- (x + 2)$ .

Tap for more steps...

Step 1.2.1.1

Apply the distributive property.

$- x - 1 \cdot 2 \geq 0$

Step 1.2.1.2

Multiply $- 1$ by $2$ .

$- x - 2 \geq 0$

Step 1.2.2

Add $2$ to both sides of the inequality.

$- x \geq 2$

Step 1.2.3

Divide each term in $- x \geq 2$ by $- 1$ and simplify.

Tap for more steps...

Step 1.2.3.1

Divide each term in $- x \geq 2$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x}{- 1} \leq \frac{2}{- 1}$

Step 1.2.3.2

Simplify the left side.

Tap for more steps...

Step 1.2.3.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} \leq \frac{2}{- 1}$

Step 1.2.3.2.2

Divide $x$ by $1$ .

$x \leq \frac{2}{- 1}$

Step 1.2.3.3

Simplify the right side.

Tap for more steps...

Step 1.2.3.3.1

Divide $2$ by $- 1$ .

$x \leq - 2$

Step 1.3

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

Interval Notation:

$(- \infty, - 2]$

Set-Builder Notation:

${x | x \leq - 2}$

Interval Notation:

$(- \infty, - 2]$

Set-Builder Notation:

${x | x \leq - 2}$

Step 2

To find the radical expression end point, substitute the $x$ value $- 2$ , which is the least value in the domain, into $f (x) = - \sqrt{- (x + 2)} - 3$ .

Tap for more steps...

Step 2.1

Replace the variable $x$ with $- 2$ in the expression.

$f (- 2) = - \sqrt{- ((- 2) + 2)} - 3$

Step 2.2

Simplify the result.

Tap for more steps...

Step 2.2.1

Simplify each term.

Tap for more steps...

Step 2.2.1.1

Add $- 2$ and $2$ .

$f (- 2) = - \sqrt{- 0} - 3$

Step 2.2.1.2

Multiply $- 1$ by $0$ .

$f (- 2) = - \sqrt{0} - 3$

Step 2.2.1.3

Rewrite $0$ as $0^{2}$ .

$f (- 2) = - \sqrt{0^{2}} - 3$

Step 2.2.1.4

Pull terms out from under the radical, assuming positive real numbers.

$f (- 2) = - 0 - 3$

Step 2.2.1.5

Multiply $- 1$ by $0$ .

$f (- 2) = 0 - 3$

Step 2.2.2

Subtract $3$ from $0$ .

$f (- 2) = - 3$

Step 2.2.3

The final answer is $- 3$ .

$- 3$

Step 3

The radical expression end point is $(- 2, - 3)$ .

$(- 2, - 3)$

Step 4

The square root can be graphed using the points around the vertex $(- 2, - 3),$

$\begin{array}{c} x & y \\ - 2 & - 3 \end{array}$

Step 5