Algebra Examples

$y = \sqrt{- x - 3}$

Step 1

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

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Step 1.1

Set the radicand in $\sqrt{- x - 3}$ greater than or equal to $0$ to find where the expression is defined.

$- x - 3 \geq 0$

Step 1.2

Solve for $x$ .

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Step 1.2.1

Add $3$ to both sides of the inequality.

$- x \geq 3$

Step 1.2.2

Divide each term in $- x \geq 3$ by $- 1$ and simplify.

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Step 1.2.2.1

Divide each term in $- x \geq 3$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x}{- 1} \leq \frac{3}{- 1}$

Step 1.2.2.2

Simplify the left side.

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Step 1.2.2.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} \leq \frac{3}{- 1}$

Step 1.2.2.2.2

Divide $x$ by $1$ .

$x \leq \frac{3}{- 1}$

Step 1.2.2.3

Simplify the right side.

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Step 1.2.2.3.1

Divide $3$ by $- 1$ .

$x \leq - 3$

Step 1.3

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

Interval Notation:

$(- \infty, - 3]$

Set-Builder Notation:

${x | x \leq - 3}$

Interval Notation:

$(- \infty, - 3]$

Set-Builder Notation:

${x | x \leq - 3}$

Step 2

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

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Step 2.1

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

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

Step 2.2

Simplify the result.

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Step 2.2.1

Multiply $- 1$ by $- 3$ .

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

Step 2.2.2

Subtract $3$ from $3$ .

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

Step 2.2.3

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

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

Step 2.2.4

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

$f (- 3) = 0$

Step 2.2.5

The final answer is $0$ .

$0$

Step 3

The radical expression end point is $(- 3, 0)$ .

$(- 3, 0)$

Step 4

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.

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Step 4.1

Substitute the $x$ value $- 5$ into $f (x) = \sqrt{- x - 3}$ . In this case, the point is $(- 5, \sqrt{2})$ .

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Step 4.1.1

Replace the variable $x$ with $- 5$ in the expression.

$f (- 5) = \sqrt{- (- 5) - 3}$

Step 4.1.2

Simplify the result.

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Step 4.1.2.1

Multiply $- 1$ by $- 5$ .

$f (- 5) = \sqrt{5 - 3}$

Step 4.1.2.2

Subtract $3$ from $5$ .

$f (- 5) = \sqrt{2}$

Step 4.1.2.3

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

$y = \sqrt{2}$

Step 4.2

Substitute the $x$ value $- 4$ into $f (x) = \sqrt{- x - 3}$ . In this case, the point is $(- 4, 1)$ .

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Step 4.2.1

Replace the variable $x$ with $- 4$ in the expression.

$f (- 4) = \sqrt{- (- 4) - 3}$

Step 4.2.2

Simplify the result.

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Step 4.2.2.1

Multiply $- 1$ by $- 4$ .

$f (- 4) = \sqrt{4 - 3}$

Step 4.2.2.2

Subtract $3$ from $4$ .

$f (- 4) = \sqrt{1}$

Step 4.2.2.3

Any root of $1$ is $1$ .

$f (- 4) = 1$

Step 4.2.2.4

The final answer is $1$ .

$y = 1$

Step 4.3

The square root can be graphed using the points around the vertex $(- 3, 0), (- 5, 1.41), (- 4, 1)$

$\begin{array}{c} x & y \\ - 5 & 1.41 \\ - 4 & 1 \\ - 3 & 0 \end{array}$

Step 5