Algebra Examples

$y = \sqrt{4 + x}$

Step 1

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

$4 + x \geq 0$

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 $f (x) = \sqrt{4 + x}$ .

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Step 2.1

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

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

Step 2.2

Simplify the result.

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Step 2.2.1

Remove parentheses.

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

Step 2.2.2

Subtract $4$ from $4$ .

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

Step 2.2.3

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

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

Step 2.2.4

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

$f (- 4) = 0$

Step 2.2.5

The final answer is $0$ .

$0$

Step 3

The radical expression end point is $(- 4, 0)$ .

$(- 4, 0)$

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

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Step 4.1.1

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

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

Step 4.1.2

Simplify the result.

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Step 4.1.2.1

Remove parentheses.

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

Step 4.1.2.2

Subtract $3$ from $4$ .

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

Step 4.1.2.3

Any root of $1$ is $1$ .

$f (- 3) = 1$

Step 4.1.2.4

The final answer is $1$ .

$y = 1$

Step 4.2

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

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Step 4.2.1

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

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

Step 4.2.2

Simplify the result.

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Step 4.2.2.1

Remove parentheses.

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

Step 4.2.2.2

Subtract $2$ from $4$ .

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

Step 4.2.2.3

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

$y = \sqrt{2}$

Step 4.3

The square root can be graphed using the points around the vertex $(- 4, 0), (- 3, 1), (- 2, 1.41)$

$\begin{array}{c} x & y \\ - 4 & 0 \\ - 3 & 1 \\ - 2 & 1.41 \end{array}$

Step 5