Algebra Examples

$f (x) = \frac{1}{2} \sqrt{x + 4} - 1$

Step 1

Find the domain for $y = \frac{1}{2} \cdot \sqrt{x + 4} - 1$ 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 $\sqrt{x + 4}$ greater than or equal to $0$ to find where the expression is defined.

$x + 4 \geq 0$

Step 1.2

Subtract $4$ from both sides of the inequality.

$x \geq - 4$

Step 1.3

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

Interval Notation:

$[- 4, \infty)$

Set-Builder Notation:

${x | x \geq - 4}$

Interval Notation:

$[- 4, \infty)$

Set-Builder Notation:

${x | x \geq - 4}$

Step 2

To find the radical expression end point, substitute the $x$ value $- 4$ , which is the least value in the domain, into $f (x) = \frac{\sqrt{x + 4} - 2}{2}$ .

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Step 2.1

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

$f (- 4) = \frac{\sqrt{(- 4) + 4} - 2}{2}$

Step 2.2

Simplify the result.

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Step 2.2.1

Simplify the numerator.

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Step 2.2.1.1

Add $- 4$ and $4$ .

$f (- 4) = \frac{\sqrt{0} - 2}{2}$

Step 2.2.1.2

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

$f (- 4) = \frac{\sqrt{0^{2}} - 2}{2}$

Step 2.2.1.3

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

$f (- 4) = \frac{0 - 2}{2}$

Step 2.2.1.4

Subtract $2$ from $0$ .

$f (- 4) = \frac{- 2}{2}$

Step 2.2.2

Divide $- 2$ by $2$ .

$f (- 4) = - 1$

Step 2.2.3

The final answer is $- 1$ .

$- 1$

Step 3

The radical expression end point is $(- 4, - 1)$ .

$(- 4, - 1)$

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 $- 3$ into $f (x) = \frac{\sqrt{x + 4} - 2}{2}$ . In this case, the point is $(- 3, - \frac{1}{2})$ .

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Step 4.1.1

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

$f (- 3) = \frac{\sqrt{(- 3) + 4} - 2}{2}$

Step 4.1.2

Simplify the result.

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Step 4.1.2.1

Simplify the numerator.

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Step 4.1.2.1.1

Add $- 3$ and $4$ .

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

Step 4.1.2.1.2

Any root of $1$ is $1$ .

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

Step 4.1.2.1.3

Subtract $2$ from $1$ .

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

Step 4.1.2.2

Move the negative in front of the fraction.

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

Step 4.1.2.3

The final answer is $- \frac{1}{2}$ .

$y = - \frac{1}{2}$

Step 4.2

Substitute the $x$ value $- 2$ into $f (x) = \frac{\sqrt{x + 4} - 2}{2}$ . In this case, the point is $(- 2, \frac{\sqrt{2} - 2}{2})$ .

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Step 4.2.1

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

$f (- 2) = \frac{\sqrt{(- 2) + 4} - 2}{2}$

Step 4.2.2

Simplify the result.

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Step 4.2.2.1

Add $- 2$ and $4$ .

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

Step 4.2.2.2

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

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

Step 4.3

The square root can be graphed using the points around the vertex $(- 4, - 1), (- 3, - 0.5), (- 2, - 0.29)$

$\begin{array}{c} x & y \\ - 4 & - 1 \\ - 3 & - 0.5 \\ - 2 & - 0.29 \end{array}$

Step 5