Trigonometry Examples

$\frac{4}{\sqrt{2 x - 5}}$

Step 1

Find the domain for $y = \frac{4}{\sqrt{2 x - 5}}$ 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 $\sqrt{2 x - 5}$ greater than or equal to $0$ to find where the expression is defined.

$2 x - 5 \geq 0$

Step 1.2

Solve for $x$ .

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Step 1.2.1

Add $5$ to both sides of the inequality.

$2 x \geq 5$

Step 1.2.2

Divide each term in $2 x \geq 5$ by $2$ and simplify.

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Step 1.2.2.1

Divide each term in $2 x \geq 5$ by $2$ .

$\frac{2 x}{2} \geq \frac{5}{2}$

Step 1.2.2.2

Simplify the left side.

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Step 1.2.2.2.1

Cancel the common factor of $2$ .

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Step 1.2.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} \geq \frac{5}{2}$

Step 1.2.2.2.1.2

Divide $x$ by $1$ .

$x \geq \frac{5}{2}$

Step 1.3

Set the denominator in $\frac{4 \sqrt{2 x - 5}}{2 x - 5}$ equal to $0$ to find where the expression is undefined.

$2 x - 5 = 0$

Step 1.4

Solve for $x$ .

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Step 1.4.1

Add $5$ to both sides of the equation.

$2 x = 5$

Step 1.4.2

Divide each term in $2 x = 5$ by $2$ and simplify.

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Step 1.4.2.1

Divide each term in $2 x = 5$ by $2$ .

$\frac{2 x}{2} = \frac{5}{2}$

Step 1.4.2.2

Simplify the left side.

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Step 1.4.2.2.1

Cancel the common factor of $2$ .

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Step 1.4.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{5}{2}$

Step 1.4.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{5}{2}$

Step 1.5

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

Interval Notation:

$(\frac{5}{2}, \infty)$

Set-Builder Notation:

${x | x > \frac{5}{2}}$

Interval Notation:

$(\frac{5}{2}, \infty)$

Set-Builder Notation:

${x | x > \frac{5}{2}}$

Step 2

To find the radical expression end point, substitute the $x$ value $\frac{5}{2}$ , which is the least value in the domain, into $f (x) = \frac{4 \sqrt{2 x - 5}}{2 x - 5}$ .

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Step 2.1

Replace the variable $x$ with $\frac{5}{2}$ in the expression.

$f (\frac{5}{2}) = \frac{4 \sqrt{2 (\frac{5}{2}) - 5}}{2 (\frac{5}{2}) - 5}$

Step 2.2

Cancel the common factor of $2$ .

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Step 2.2.1

Cancel the common factor.

$f (\frac{5}{2}) = \frac{4 \sqrt{2 (\frac{5}{2}) - 5}}{2 (\frac{5}{2}) - 5}$

Step 2.2.2

Rewrite the expression.

$f (\frac{5}{2}) = \frac{4 \sqrt{2 (\frac{5}{2}) - 5}}{5 - 5}$

Step 2.3

Subtract $5$ from $5$ .

$f (\frac{5}{2}) = \frac{4 \sqrt{2 (\frac{5}{2}) - 5}}{0}$

Step 2.4

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

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

Undefined

Step 3

The radical expression end point is $(\frac{5}{2}, Undefined)$ .

$(\frac{5}{2}, Undefined)$

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 $f (x) = \frac{4 \sqrt{2 x - 5}}{2 x - 5}$ . In this case, the point is $(3, 4)$ .

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Step 4.1.1

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

$f (3) = \frac{4 \sqrt{2 (3) - 5}}{2 (3) - 5}$

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

Multiply $2$ by $3$ .

$f (3) = \frac{4 \sqrt{6 - 5}}{2 (3) - 5}$

Step 4.1.2.1.2

Subtract $5$ from $6$ .

$f (3) = \frac{4 \sqrt{1}}{2 (3) - 5}$

Step 4.1.2.1.3

Any root of $1$ is $1$ .

$f (3) = \frac{4 \cdot 1}{2 (3) - 5}$

Step 4.1.2.2

Simplify the denominator.

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Step 4.1.2.2.1

Multiply $2$ by $3$ .

$f (3) = \frac{4 \cdot 1}{6 - 5}$

Step 4.1.2.2.2

Subtract $5$ from $6$ .

$f (3) = \frac{4 \cdot 1}{1}$

Step 4.1.2.3

Simplify the expression.

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Step 4.1.2.3.1

Multiply $4$ by $1$ .

$f (3) = \frac{4}{1}$

Step 4.1.2.3.2

Divide $4$ by $1$ .

$f (3) = 4$

Step 4.1.2.4

The final answer is $4$ .

$y = 4$

Step 4.2

Substitute the $x$ value $4$ into $f (x) = \frac{4 \sqrt{2 x - 5}}{2 x - 5}$ . In this case, the point is $(4, \frac{4 \sqrt{3}}{3})$ .

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Step 4.2.1

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

$f (4) = \frac{4 \sqrt{2 (4) - 5}}{2 (4) - 5}$

Step 4.2.2

Simplify the result.

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Step 4.2.2.1

Simplify the numerator.

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Step 4.2.2.1.1

Multiply $2$ by $4$ .

$f (4) = \frac{4 \sqrt{8 - 5}}{2 (4) - 5}$

Step 4.2.2.1.2

Subtract $5$ from $8$ .

$f (4) = \frac{4 \sqrt{3}}{2 (4) - 5}$

Step 4.2.2.2

Simplify the denominator.

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Step 4.2.2.2.1

Multiply $2$ by $4$ .

$f (4) = \frac{4 \sqrt{3}}{8 - 5}$

Step 4.2.2.2.2

Subtract $5$ from $8$ .

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

Step 4.2.2.3

The final answer is $\frac{4 \sqrt{3}}{3}$ .

$y = \frac{4 \sqrt{3}}{3}$

Step 4.3

Substitute the $x$ value $5$ into $f (x) = \frac{4 \sqrt{2 x - 5}}{2 x - 5}$ . In this case, the point is $(5, \frac{4 \sqrt{5}}{5})$ .

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Step 4.3.1

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

$f (5) = \frac{4 \sqrt{2 (5) - 5}}{2 (5) - 5}$

Step 4.3.2

Simplify the result.

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Step 4.3.2.1

Simplify the numerator.

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Step 4.3.2.1.1

Multiply $2$ by $5$ .

$f (5) = \frac{4 \sqrt{10 - 5}}{2 (5) - 5}$

Step 4.3.2.1.2

Subtract $5$ from $10$ .

$f (5) = \frac{4 \sqrt{5}}{2 (5) - 5}$

Step 4.3.2.2

Simplify the denominator.

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Step 4.3.2.2.1

Multiply $2$ by $5$ .

$f (5) = \frac{4 \sqrt{5}}{10 - 5}$

Step 4.3.2.2.2

Subtract $5$ from $10$ .

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

Step 4.3.2.3

The final answer is $\frac{4 \sqrt{5}}{5}$ .

$y = \frac{4 \sqrt{5}}{5}$

Step 4.4

The square root can be graphed using the points around the vertex $(2.5, Undefined), (3, 4), (4, 2.31), (5, 1.79)$

$\begin{array}{c} x & y \\ 2.5 & Undefined \\ 3 & 4 \\ 4 & 2.31 \\ 5 & 1.79 \end{array}$

Step 5