Graph y = square root of -x-1-2

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

Find the domain for $y = \sqrt{- x - 1} - 2$ so that a list of $x$ values can be picked to find a list of points, which will help graphing the radical.

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

$- x - 1 \geq 0$

Solve for $x$ .

Add $1$ to both sides of the inequality.

$- x \geq 1$

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

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

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

Simplify the left side.

Dividing two negative values results in a positive value.

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

Divide $x$ by $1$ .

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

Simplify the right side.

Divide $1$ by $- 1$ .

$x \leq - 1$

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

Interval Notation:

$(- \infty, - 1]$

Set-Builder Notation:

${x | x \leq - 1}$

Interval Notation:

$(- \infty, - 1]$

Set-Builder Notation:

${x | x \leq - 1}$

To find the radical expression end point, substitute the $x$ value $- 1$ , which is the least value in the domain, into $f (x) = \sqrt{- x - 1} - 2$ .

Replace the variable $x$ with $- 1$ in the expression.

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

Simplify the result.

Simplify each term.

Multiply $- 1$ by $- 1$ .

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

Subtract $1$ from $1$ .

$f (- 1) = \sqrt{0} - 2$

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

$f (- 1) = \sqrt{0^{2}} - 2$

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

$f (- 1) = 0 - 2$

Subtract $2$ from $0$ .

$f (- 1) = - 2$

The final answer is $- 2$ .

$- 2$

The radical expression end point is $(- 1, - 2)$ .

$(- 1, - 2)$

Select a few $x$ values from the domain. It would be more useful to select the values so that they are next to the $x$ value of the radical expression end point.

Substitute the $x$ value $- 3$ into $f (x) = \sqrt{- x - 1} - 2$ . In this case, the point is $(- 3, \sqrt{2} - 2)$ .

Replace the variable $x$ with $- 3$ in the expression.

$f (- 3) = \sqrt{- (- 3) - 1} - 2$

Simplify the result.

Simplify each term.

Multiply $- 1$ by $- 3$ .

$f (- 3) = \sqrt{3 - 1} - 2$

Subtract $1$ from $3$ .

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

The final answer is $\sqrt{2} - 2$ .

$y = \sqrt{2} - 2$

Substitute the $x$ value $- 2$ into $f (x) = \sqrt{- x - 1} - 2$ . In this case, the point is $(- 2, - 1)$ .

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

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

Simplify the result.

Simplify each term.

Multiply $- 1$ by $- 2$ .

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

Subtract $1$ from $2$ .

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

Any root of $1$ is $1$ .

$f (- 2) = 1 - 2$

Subtract $2$ from $1$ .

$f (- 2) = - 1$

The final answer is $- 1$ .

$y = - 1$

The square root can be graphed using the points around the vertex $(- 1, - 2), (- 3, - 0.59), (- 2, - 1)$

$\begin{array}{c} x & y \\ - 3 & - 0.59 \\ - 2 & - 1 \\ - 1 & - 2 \end{array}$

Graph y = square root of -x-1-2

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