Algebra Examples

$\sqrt{\frac{1}{3} x}$

Step 1

Find the domain for $y = \sqrt{\frac{1}{3} x}$ 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 $\sqrt{3 x}$ greater than or equal to $0$ to find where the expression is defined.

$3 x \geq 0$

Step 1.2

Divide each term in $3 x \geq 0$ by $3$ and simplify.

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Step 1.2.1

Divide each term in $3 x \geq 0$ by $3$ .

$\frac{3 x}{3} \geq \frac{0}{3}$

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Cancel the common factor of $3$ .

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Step 1.2.2.1.1

Cancel the common factor.

$\frac{3 x}{3} \geq \frac{0}{3}$

Step 1.2.2.1.2

Divide $x$ by $1$ .

$x \geq \frac{0}{3}$

Step 1.2.3

Simplify the right side.

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Step 1.2.3.1

Divide $0$ by $3$ .

$x \geq 0$

Step 1.3

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

Interval Notation:

$[0, \infty)$

Set-Builder Notation:

${x | x \geq 0}$

Interval Notation:

$[0, \infty)$

Set-Builder Notation:

${x | x \geq 0}$

Step 2

To find the radical expression end point, substitute the $x$ value $0$ , which is the least value in the domain, into $f (x) = \frac{\sqrt{3 x}}{3}$ .

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Step 2.1

Replace the variable $x$ with $0$ in the expression.

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

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

Multiply $3$ by $0$ .

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

Step 2.2.1.2

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

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

Step 2.2.1.3

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

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

Step 2.2.2

Divide $0$ by $3$ .

$f (0) = 0$

Step 2.2.3

The final answer is $0$ .

$0$

Step 3

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

$(0, 0)$

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

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Step 4.1.1

Replace the variable $x$ with $1$ in the expression.

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

Step 4.1.2

Simplify the result.

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Step 4.1.2.1

Multiply $3$ by $1$ .

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

Step 4.1.2.2

The final answer is $\frac{\sqrt{3}}{3}$ .

$y = \frac{\sqrt{3}}{3}$

Step 4.2

Substitute the $x$ value $2$ into $f (x) = \frac{\sqrt{3 x}}{3}$ . In this case, the point is $(2, \frac{\sqrt{6}}{3})$ .

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Step 4.2.1

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

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

Step 4.2.2

Simplify the result.

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Step 4.2.2.1

Multiply $3$ by $2$ .

$f (2) = \frac{\sqrt{6}}{3}$

Step 4.2.2.2

The final answer is $\frac{\sqrt{6}}{3}$ .

$y = \frac{\sqrt{6}}{3}$

Step 4.3

The square root can be graphed using the points around the vertex $(0, 0), (1, 0.58), (2, 0.82)$

$\begin{array}{c} x & y \\ 0 & 0 \\ 1 & 0.58 \\ 2 & 0.82 \end{array}$

Step 5