Pre-Algebra Examples

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

Step 1

Find the domain for $y = \sqrt{x - 4} + 4$ 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

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

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Step 2.1

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

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

Step 2.2

Simplify the result.

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Step 2.2.1

Simplify each term.

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Step 2.2.1.1

Subtract $4$ from $4$ .

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

Step 2.2.1.2

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

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

Step 2.2.1.3

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

$f (4) = 0 + 4$

Step 2.2.2

Add $0$ and $4$ .

$f (4) = 4$

Step 2.2.3

The final answer is $4$ .

$4$

Step 3

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

$(4, 4)$

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 $f (x) = \sqrt{x - 4} + 4$ . In this case, the point is $(5, 5)$ .

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Step 4.1.1

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

$f (5) = \sqrt{(5) - 4} + 4$

Step 4.1.2

Simplify the result.

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Step 4.1.2.1

Simplify each term.

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Step 4.1.2.1.1

Subtract $4$ from $5$ .

$f (5) = \sqrt{1} + 4$

Step 4.1.2.1.2

Any root of $1$ is $1$ .

$f (5) = 1 + 4$

Step 4.1.2.2

Add $1$ and $4$ .

$f (5) = 5$

Step 4.1.2.3

The final answer is $5$ .

$y = 5$

Step 4.2

Substitute the $x$ value $6$ into $f (x) = \sqrt{x - 4} + 4$ . In this case, the point is $(6, \sqrt{2} + 4)$ .

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Step 4.2.1

Replace the variable $x$ with $6$ in the expression.

$f (6) = \sqrt{(6) - 4} + 4$

Step 4.2.2

Simplify the result.

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Step 4.2.2.1

Subtract $4$ from $6$ .

$f (6) = \sqrt{2} + 4$

Step 4.2.2.2

The final answer is $\sqrt{2} + 4$ .

$y = \sqrt{2} + 4$

Step 4.3

The square root can be graphed using the points around the vertex $(4, 4), (5, 5), (6, 5.41)$

$\begin{array}{c} x & y \\ 4 & 4 \\ 5 & 5 \\ 6 & 5.41 \end{array}$

Step 5