Graph f(x)=1/3 square root of x-2

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

Step 1

Find the domain for $y = \frac{1}{3} \cdot \sqrt{x - 2}$ so that a list of $x$ values can be picked to find a list of points, which will help graphing the radical.

Set the radicand in $\sqrt{x - 2}$ greater than or equal to $0$ to find where the expression is defined.

$x - 2 \geq 0$

Add $2$ to both sides of the inequality.

$x \geq 2$

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

Interval Notation:

$[2, \infty)$

Set-Builder Notation:

${x | x \geq 2}$

Interval Notation:

$[2, \infty)$

Set-Builder Notation:

${x | x \geq 2}$

Step 2

To find the radical expression end point, substitute the $x$ value $2$ , which is the least value in the domain, into $f (x) = \frac{\sqrt{x - 2}}{3}$ .

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

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

Simplify the result.

Simplify the numerator.

Subtract $2$ from $2$ .

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

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

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

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

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

Divide $0$ by $3$ .

$f (2) = 0$

The final answer is $0$ .

$0$

Step 3

The radical expression end point is $(2, 0)$ .

$(2, 0)$

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.

Substitute the $x$ value $3$ into $f (x) = \frac{\sqrt{x - 2}}{3}$ . In this case, the point is $(3, \frac{1}{3})$ .

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

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

Simplify the result.

Simplify the numerator.

Subtract $2$ from $3$ .

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

Any root of $1$ is $1$ .

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

The final answer is $\frac{1}{3}$ .

$y = \frac{1}{3}$

Substitute the $x$ value $4$ into $f (x) = \frac{\sqrt{x - 2}}{3}$ . In this case, the point is $(4, \frac{\sqrt{2}}{3})$ .

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

$f (4) = \frac{\sqrt{(4) - 2}}{3}$

Simplify the result.

Subtract $2$ from $4$ .

$f (4) = \frac{\sqrt{2}}{3}$

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

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

The square root can be graphed using the points around the vertex $(2, 0), (3, 0.33), (4, 0.47)$

$\begin{array}{c} x & y \\ 2 & 0 \\ 3 & 0.33 \\ 4 & 0.47 \end{array}$

Step 5

Graph f(x)=1/3 square root of x-2

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