Algebra Examples

$F (x) = \frac{1}{2} \sqrt{x} + 3$

Step 1

Find the domain for $y = \frac{1}{2} \cdot \sqrt{x} + 3$ 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 $\sqrt{x}$ greater than or equal to $0$ to find where the expression is defined.

$x \geq 0$

Step 1.2

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

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Step 2.1

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

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

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

Simplify the numerator.

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Step 2.2.1.1.1

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

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

Step 2.2.1.1.2

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

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

Step 2.2.1.2

Divide $0$ by $2$ .

$f (0) = 0 + 3$

Step 2.2.2

Add $0$ and $3$ .

$f (0) = 3$

Step 2.2.3

The final answer is $3$ .

$3$

Step 3

The radical expression end point is $(0, 3)$ .

$(0, 3)$

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 $f (x) = \frac{\sqrt{x}}{2} + 3$ . In this case, the point is $(1, \frac{7}{2})$ .

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Step 4.1.1

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

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

Step 4.1.2

Simplify the result.

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Step 4.1.2.1

Any root of $1$ is $1$ .

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

Step 4.1.2.2

To write $3$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 4.1.2.3

Combine $3$ and $\frac{2}{2}$ .

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

Step 4.1.2.4

Combine the numerators over the common denominator.

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

Step 4.1.2.5

Simplify the numerator.

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Step 4.1.2.5.1

Multiply $3$ by $2$ .

$f (1) = \frac{1 + 6}{2}$

Step 4.1.2.5.2

Add $1$ and $6$ .

$f (1) = \frac{7}{2}$

Step 4.1.2.6

The final answer is $\frac{7}{2}$ .

$y = \frac{7}{2}$

Step 4.2

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

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Step 4.2.1

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

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

Step 4.2.2

The final answer is $\frac{\sqrt{2}}{2} + 3$ .

$y = \frac{\sqrt{2}}{2} + 3$

Step 4.3

The square root can be graphed using the points around the vertex $(0, 3), (1, 3.5), (2, 3.71)$

$\begin{array}{c} x & y \\ 0 & 3 \\ 1 & 3.5 \\ 2 & 3.71 \end{array}$

Step 5