Trigonometry Examples

$f (x) = \sqrt{\frac{2 - 81 x}{3 x + 8}}$

Step 1

Find the domain for $y = \sqrt{\frac{2 - 81 x}{3 x + 8}}$ 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 $\sqrt{(2 - 81 x) (3 x + 8)}$ greater than or equal to $0$ to find where the expression is defined.

$(2 - 81 x) (3 x + 8) \geq 0$

Step 1.2

Solve for $x$ .

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Step 1.2.1

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$2 - 81 x = 0$

$3 x + 8 = 0$

Step 1.2.2

Set $2 - 81 x$ equal to $0$ and solve for $x$ .

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Step 1.2.2.1

Set $2 - 81 x$ equal to $0$ .

$2 - 81 x = 0$

Step 1.2.2.2

Solve $2 - 81 x = 0$ for $x$ .

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Step 1.2.2.2.1

Subtract $2$ from both sides of the equation.

$- 81 x = - 2$

Step 1.2.2.2.2

Divide each term in $- 81 x = - 2$ by $- 81$ and simplify.

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Step 1.2.2.2.2.1

Divide each term in $- 81 x = - 2$ by $- 81$ .

$\frac{- 81 x}{- 81} = \frac{- 2}{- 81}$

Step 1.2.2.2.2.2

Simplify the left side.

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Step 1.2.2.2.2.2.1

Cancel the common factor of $- 81$ .

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Step 1.2.2.2.2.2.1.1

Cancel the common factor.

$\frac{- 81 x}{- 81} = \frac{- 2}{- 81}$

Step 1.2.2.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 2}{- 81}$

Step 1.2.2.2.2.3

Simplify the right side.

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Step 1.2.2.2.2.3.1

Dividing two negative values results in a positive value.

$x = \frac{2}{81}$

Step 1.2.3

Set $3 x + 8$ equal to $0$ and solve for $x$ .

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Step 1.2.3.1

Set $3 x + 8$ equal to $0$ .

$3 x + 8 = 0$

Step 1.2.3.2

Solve $3 x + 8 = 0$ for $x$ .

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Step 1.2.3.2.1

Subtract $8$ from both sides of the equation.

$3 x = - 8$

Step 1.2.3.2.2

Divide each term in $3 x = - 8$ by $3$ and simplify.

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Step 1.2.3.2.2.1

Divide each term in $3 x = - 8$ by $3$ .

$\frac{3 x}{3} = \frac{- 8}{3}$

Step 1.2.3.2.2.2

Simplify the left side.

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Step 1.2.3.2.2.2.1

Cancel the common factor of $3$ .

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Step 1.2.3.2.2.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{- 8}{3}$

Step 1.2.3.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 8}{3}$

Step 1.2.3.2.2.3

Simplify the right side.

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Step 1.2.3.2.2.3.1

Move the negative in front of the fraction.

$x = - \frac{8}{3}$

Step 1.2.4

The final solution is all the values that make $(2 - 81 x) (3 x + 8) \geq 0$ true.

$x = \frac{2}{81}, - \frac{8}{3}$

Step 1.2.5

Use each root to create test intervals.

$x < - \frac{8}{3}$

$- \frac{8}{3} < x < \frac{2}{81}$

$x > \frac{2}{81}$

Step 1.2.6

Choose a test value from each interval and plug this value into the original inequality to determine which intervals satisfy the inequality.

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Step 1.2.6.1

Test a value on the interval $x < - \frac{8}{3}$ to see if it makes the inequality true.

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Step 1.2.6.1.1

Choose a value on the interval $x < - \frac{8}{3}$ and see if this value makes the original inequality true.

$x = - 5$

Step 1.2.6.1.2

Replace $x$ with $- 5$ in the original inequality.

$(2 - 81 \cdot - 5) (3 (- 5) + 8) \geq 0$

Step 1.2.6.1.3

The left side $- 2849$ is less than the right side $0$ , which means that the given statement is false.

False

Step 1.2.6.2

Test a value on the interval $- \frac{8}{3} < x < \frac{2}{81}$ to see if it makes the inequality true.

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Step 1.2.6.2.1

Choose a value on the interval $- \frac{8}{3} < x < \frac{2}{81}$ and see if this value makes the original inequality true.

$x = 0$

Step 1.2.6.2.2

Replace $x$ with $0$ in the original inequality.

$(2 - 81 \cdot 0) (3 (0) + 8) \geq 0$

Step 1.2.6.2.3

The left side $16$ is greater than the right side $0$ , which means that the given statement is always true.

True

Step 1.2.6.3

Test a value on the interval $x > \frac{2}{81}$ to see if it makes the inequality true.

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Step 1.2.6.3.1

Choose a value on the interval $x > \frac{2}{81}$ and see if this value makes the original inequality true.

$x = 3$

Step 1.2.6.3.2

Replace $x$ with $3$ in the original inequality.

$(2 - 81 \cdot 3) (3 (3) + 8) \geq 0$

Step 1.2.6.3.3

The left side $- 4097$ is less than the right side $0$ , which means that the given statement is false.

False

Step 1.2.6.4

Compare the intervals to determine which ones satisfy the original inequality.

$x < - \frac{8}{3}$ False

$- \frac{8}{3} < x < \frac{2}{81}$ True

$x > \frac{2}{81}$ False

$x < - \frac{8}{3}$ False

$- \frac{8}{3} < x < \frac{2}{81}$ True

$x > \frac{2}{81}$ False

Step 1.2.7

The solution consists of all of the true intervals.

$- \frac{8}{3} \leq x \leq \frac{2}{81}$

Step 1.3

Set the denominator in $\frac{\sqrt{(2 - 81 x) (3 x + 8)}}{3 x + 8}$ equal to $0$ to find where the expression is undefined.

$3 x + 8 = 0$

Step 1.4

Solve for $x$ .

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Step 1.4.1

Subtract $8$ from both sides of the equation.

$3 x = - 8$

Step 1.4.2

Divide each term in $3 x = - 8$ by $3$ and simplify.

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Step 1.4.2.1

Divide each term in $3 x = - 8$ by $3$ .

$\frac{3 x}{3} = \frac{- 8}{3}$

Step 1.4.2.2

Simplify the left side.

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Step 1.4.2.2.1

Cancel the common factor of $3$ .

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Step 1.4.2.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{- 8}{3}$

Step 1.4.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 8}{3}$

Step 1.4.2.3

Simplify the right side.

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Step 1.4.2.3.1

Move the negative in front of the fraction.

$x = - \frac{8}{3}$

Step 1.5

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

Interval Notation:

$(- \frac{8}{3}, \frac{2}{81}]$

Set-Builder Notation:

${x | - \frac{8}{3} < x \leq \frac{2}{81}}$

Interval Notation:

$(- \frac{8}{3}, \frac{2}{81}]$

Set-Builder Notation:

${x | - \frac{8}{3} < x \leq \frac{2}{81}}$

Step 2

To find the end points, substitute the bounds of the $x$ values from the domain into $f (x) = \frac{\sqrt{(2 - 81 x) (3 x + 8)}}{3 x + 8}$ .

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Step 2.1

Replace the variable $x$ with $- \frac{8}{3}$ in the expression.

$f (- \frac{8}{3}) = \frac{\sqrt{(2 - 81 (- \frac{8}{3})) (3 (- \frac{8}{3}) + 8)}}{3 (- \frac{8}{3}) + 8}$

Step 2.2

Cancel the common factor of $3$ .

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Step 2.2.1

Move the leading negative in $- \frac{8}{3}$ into the numerator.

$f (- \frac{8}{3}) = \frac{\sqrt{(2 - 81 (- \frac{8}{3})) (3 (- \frac{8}{3}) + 8)}}{3 (\frac{- 8}{3}) + 8}$

Step 2.2.2

Cancel the common factor.

$f (- \frac{8}{3}) = \frac{\sqrt{(2 - 81 (- \frac{8}{3})) (3 (- \frac{8}{3}) + 8)}}{3 (\frac{- 8}{3}) + 8}$

Step 2.2.3

Rewrite the expression.

$f (- \frac{8}{3}) = \frac{\sqrt{(2 - 81 (- \frac{8}{3})) (3 (- \frac{8}{3}) + 8)}}{- 8 + 8}$

Step 2.3

Add $- 8$ and $8$ .

$f (- \frac{8}{3}) = \frac{\sqrt{(2 - 81 (- \frac{8}{3})) (3 (- \frac{8}{3}) + 8)}}{0}$

Step 2.4

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

$f (- \frac{8}{3}) = Undefined$

Step 2.5

Replace the variable $x$ with $\frac{2}{81}$ in the expression.

$f (\frac{2}{81}) = \frac{\sqrt{(2 - 81 (\frac{2}{81})) (3 (\frac{2}{81}) + 8)}}{3 (\frac{2}{81}) + 8}$

Step 2.6

Simplify the result.

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Step 2.6.1

Simplify the numerator.

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Step 2.6.1.1

Cancel the common factor of $81$ .

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Step 2.6.1.1.1

Factor $81$ out of $- 81$ .

$f (\frac{2}{81}) = \frac{\sqrt{(2 + 81 (- 1) (\frac{2}{81})) (3 (\frac{2}{81}) + 8)}}{3 (\frac{2}{81}) + 8}$

Step 2.6.1.1.2

Cancel the common factor.

$f (\frac{2}{81}) = \frac{\sqrt{(2 + 81 \cdot (- 1 (\frac{2}{81}))) (3 (\frac{2}{81}) + 8)}}{3 (\frac{2}{81}) + 8}$

Step 2.6.1.1.3

Rewrite the expression.

$f (\frac{2}{81}) = \frac{\sqrt{(2 - 1 \cdot 2) (3 (\frac{2}{81}) + 8)}}{3 (\frac{2}{81}) + 8}$

Step 2.6.1.2

Multiply $- 1$ by $2$ .

$f (\frac{2}{81}) = \frac{\sqrt{(2 - 2) (3 (\frac{2}{81}) + 8)}}{3 (\frac{2}{81}) + 8}$

Step 2.6.1.3

Subtract $2$ from $2$ .

$f (\frac{2}{81}) = \frac{\sqrt{0 (3 (\frac{2}{81}) + 8)}}{3 (\frac{2}{81}) + 8}$

Step 2.6.1.4

Cancel the common factor of $3$ .

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Step 2.6.1.4.1

Factor $3$ out of $81$ .

$f (\frac{2}{81}) = \frac{\sqrt{0 (3 (\frac{2}{3 (27)}) + 8)}}{3 (\frac{2}{81}) + 8}$

Step 2.6.1.4.2

Cancel the common factor.

$f (\frac{2}{81}) = \frac{\sqrt{0 (3 (\frac{2}{3 \cdot 27}) + 8)}}{3 (\frac{2}{81}) + 8}$

Step 2.6.1.4.3

Rewrite the expression.

$f (\frac{2}{81}) = \frac{\sqrt{0 (\frac{2}{27} + 8)}}{3 (\frac{2}{81}) + 8}$

Step 2.6.1.5

To write $8$ as a fraction with a common denominator, multiply by $\frac{27}{27}$ .

$f (\frac{2}{81}) = \frac{\sqrt{0 (\frac{2}{27} + 8 \cdot \frac{27}{27})}}{3 (\frac{2}{81}) + 8}$

Step 2.6.1.6

Combine $8$ and $\frac{27}{27}$ .

$f (\frac{2}{81}) = \frac{\sqrt{0 (\frac{2}{27} + \frac{8 \cdot 27}{27})}}{3 (\frac{2}{81}) + 8}$

Step 2.6.1.7

Combine the numerators over the common denominator.

$f (\frac{2}{81}) = \frac{\sqrt{0 (\frac{2 + 8 \cdot 27}{27})}}{3 (\frac{2}{81}) + 8}$

Step 2.6.1.8

Simplify the numerator.

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Step 2.6.1.8.1

Multiply $8$ by $27$ .

$f (\frac{2}{81}) = \frac{\sqrt{0 (\frac{2 + 216}{27})}}{3 (\frac{2}{81}) + 8}$

Step 2.6.1.8.2

Add $2$ and $216$ .

$f (\frac{2}{81}) = \frac{\sqrt{0 (\frac{218}{27})}}{3 (\frac{2}{81}) + 8}$

Step 2.6.1.9

Multiply $0$ by $\frac{218}{27}$ .

$f (\frac{2}{81}) = \frac{\sqrt{0}}{3 (\frac{2}{81}) + 8}$

Step 2.6.1.10

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

$f (\frac{2}{81}) = \frac{\sqrt{0^{2}}}{3 (\frac{2}{81}) + 8}$

Step 2.6.1.11

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

$f (\frac{2}{81}) = \frac{0}{3 (\frac{2}{81}) + 8}$

Step 2.6.2

Simplify the denominator.

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Step 2.6.2.1

Cancel the common factor of $3$ .

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Step 2.6.2.1.1

Factor $3$ out of $81$ .

$f (\frac{2}{81}) = \frac{0}{3 (\frac{2}{3 (27)}) + 8}$

Step 2.6.2.1.2

Cancel the common factor.

$f (\frac{2}{81}) = \frac{0}{3 (\frac{2}{3 \cdot 27}) + 8}$

Step 2.6.2.1.3

Rewrite the expression.

$f (\frac{2}{81}) = \frac{0}{\frac{2}{27} + 8}$

Step 2.6.2.2

To write $8$ as a fraction with a common denominator, multiply by $\frac{27}{27}$ .

$f (\frac{2}{81}) = \frac{0}{\frac{2}{27} + 8 \cdot \frac{27}{27}}$

Step 2.6.2.3

Combine $8$ and $\frac{27}{27}$ .

$f (\frac{2}{81}) = \frac{0}{\frac{2}{27} + \frac{8 \cdot 27}{27}}$

Step 2.6.2.4

Combine the numerators over the common denominator.

$f (\frac{2}{81}) = \frac{0}{\frac{2 + 8 \cdot 27}{27}}$

Step 2.6.2.5

Simplify the numerator.

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Step 2.6.2.5.1

Multiply $8$ by $27$ .

$f (\frac{2}{81}) = \frac{0}{\frac{2 + 216}{27}}$

Step 2.6.2.5.2

Add $2$ and $216$ .

$f (\frac{2}{81}) = \frac{0}{\frac{218}{27}}$

Step 2.6.3

Multiply the numerator by the reciprocal of the denominator.

$f (\frac{2}{81}) = 0 (\frac{27}{218})$

Step 2.6.4

Multiply $0$ by $\frac{27}{218}$ .

$f (\frac{2}{81}) = 0$

Step 2.6.5

The final answer is $0$ .

$0$

Step 3

The end points are $(- \frac{8}{3}, Undefined), (\frac{2}{81}, 0)$ .

$(- \frac{8}{3}, Undefined), (\frac{2}{81}, 0)$

Step 4

The square root can be graphed using the points around the vertex $(- \frac{8}{3}, Undefined), (\frac{2}{81}, 0),$

$\begin{array}{c} x & y \\ - \frac{8}{3} & Undefined \\ \frac{2}{81} & 0 \end{array}$

Step 5