Algebra Examples

$g (x) = \frac{1}{5} \sqrt{x - 3}$

Step 1

Find the domain for $y = \frac{1}{5} \cdot \sqrt{x - 3}$ 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

Add $3$ to both sides of the inequality.

$x \geq 3$

Step 1.3

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

Interval Notation:

$[3, \infty)$

Set-Builder Notation:

${x | x \geq 3}$

Interval Notation:

$[3, \infty)$

Set-Builder Notation:

${x | x \geq 3}$

Step 2

To find the radical expression end point, substitute the $x$ value $3$ , which is the least value in the domain, into $f (x) = \frac{\sqrt{x - 3}}{5}$ .

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Step 2.1

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

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

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

Subtract $3$ from $3$ .

$f (3) = \frac{\sqrt{0}}{5}$

Step 2.2.1.2

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

$f (3) = \frac{\sqrt{0^{2}}}{5}$

Step 2.2.1.3

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

$f (3) = \frac{0}{5}$

Step 2.2.2

Divide $0$ by $5$ .

$f (3) = 0$

Step 2.2.3

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 $4$ into $f (x) = \frac{\sqrt{x - 3}}{5}$ . In this case, the point is $(4, \frac{1}{5})$ .

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Step 4.1.1

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

$f (4) = \frac{\sqrt{(4) - 3}}{5}$

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

Subtract $3$ from $4$ .

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

Step 4.1.2.1.2

Any root of $1$ is $1$ .

$f (4) = \frac{1}{5}$

Step 4.1.2.2

The final answer is $\frac{1}{5}$ .

$y = \frac{1}{5}$

Step 4.2

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

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Step 4.2.1

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

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

Step 4.2.2

Simplify the result.

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Step 4.2.2.1

Subtract $3$ from $5$ .

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

Step 4.2.2.2

The final answer is $\frac{\sqrt{2}}{5}$ .

$y = \frac{\sqrt{2}}{5}$

Step 4.3

The square root can be graphed using the points around the vertex $(3, 0), (4, 0.2), (5, 0.28)$

$\begin{array}{c} x & y \\ 3 & 0 \\ 4 & 0.2 \\ 5 & 0.28 \end{array}$

Step 5