Trigonometry Examples

$\frac{1}{\sqrt{x - 2}}$

Step 1

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

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Step 1.1

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

$x - 2 \geq 0$

Step 1.2

Add $2$ to both sides of the inequality.

$x \geq 2$

Step 1.3

Set the denominator in $\frac{\sqrt{x - 2}}{x - 2}$ equal to $0$ to find where the expression is undefined.

$x - 2 = 0$

Step 1.4

Add $2$ to both sides of the equation.

$x = 2$

Step 1.5

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

Interval Notation:

$(2, \infty)$

Set-Builder Notation:

${x | x > 2}$

Interval Notation:

$(2, \infty)$

Set-Builder Notation:

${x | x > 2}$

Step 2

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

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Step 2.1

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

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

Step 2.2

Subtract $2$ from $2$ .

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

Step 2.3

The expression contains a division by $0$ . The expression is undefined.

$f (2) = Undefined$

Undefined

Step 3

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

$(2, Undefined)$

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

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Step 4.1.1

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

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

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 $2$ from $3$ .

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

Step 4.1.2.1.2

Any root of $1$ is $1$ .

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

Step 4.1.2.2

Simplify the expression.

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Step 4.1.2.2.1

Subtract $2$ from $3$ .

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

Step 4.1.2.2.2

Divide $1$ by $1$ .

$f (3) = 1$

Step 4.1.2.3

The final answer is $1$ .

$y = 1$

Step 4.2

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

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Step 4.2.1

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

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

Step 4.2.2

Simplify the result.

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Step 4.2.2.1

Subtract $2$ from $4$ .

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

Step 4.2.2.2

Subtract $2$ from $4$ .

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

Step 4.2.2.3

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

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

Step 4.3

The square root can be graphed using the points around the vertex $(2, Undefined), (3, 1), (4, 0.71)$

$\begin{array}{c} x & y \\ 2 & Undefined \\ 3 & 1 \\ 4 & 0.71 \end{array}$

Step 5