Trigonometry Examples

$f (x) = \sqrt{x - 5} - \log_{3} (11 - x)$

Step 1

Find the domain for $y = \sqrt{x - 5} - \log_{3} (11 - x)$ 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 argument in $\log_{3} (11 - x)$ greater than $0$ to find where the expression is defined.

$11 - x > 0$

Step 1.2

Solve for $x$ .

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Step 1.2.1

Subtract $11$ from both sides of the inequality.

$- x > - 11$

Step 1.2.2

Divide each term in $- x > - 11$ by $- 1$ and simplify.

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Step 1.2.2.1

Divide each term in $- x > - 11$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x}{- 1} < \frac{- 11}{- 1}$

Step 1.2.2.2

Simplify the left side.

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Step 1.2.2.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} < \frac{- 11}{- 1}$

Step 1.2.2.2.2

Divide $x$ by $1$ .

$x < \frac{- 11}{- 1}$

Step 1.2.2.3

Simplify the right side.

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Step 1.2.2.3.1

Divide $- 11$ by $- 1$ .

$x < 11$

Step 1.3

Set the radicand in $\sqrt{x - 5}$ greater than or equal to $0$ to find where the expression is defined.

$x - 5 \geq 0$

Step 1.4

Add $5$ to both sides of the inequality.

$x \geq 5$

Step 1.5

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

Interval Notation:

$[5, 11)$

Set-Builder Notation:

${x | 5 \leq x < 11}$

Interval Notation:

$[5, 11)$

Set-Builder Notation:

${x | 5 \leq x < 11}$

Step 2

To find the end points, substitute the bounds of the $x$ values from the domain into $f (x) = \sqrt{x - 5} - \log_{3} (11 - x)$ .

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Step 2.1

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

$f (5) = \sqrt{(5) - 5} - \log_{3} (11 - (5))$

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

Subtract $5$ from $5$ .

$f (5) = \sqrt{0} - \log_{3} (11 - (5))$

Step 2.2.1.2

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

$f (5) = \sqrt{0^{2}} - \log_{3} (11 - (5))$

Step 2.2.1.3

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

$f (5) = 0 - \log_{3} (11 - (5))$

Step 2.2.1.4

Multiply $- 1$ by $5$ .

$f (5) = 0 - \log_{3} (11 - 5)$

Step 2.2.1.5

Subtract $5$ from $11$ .

$f (5) = 0 - \log_{3} (6)$

Step 2.2.2

Subtract $\log_{3} (6)$ from $0$ .

$f (5) = - \log_{3} (6)$

Step 2.2.3

The final answer is $- \log_{3} (6)$ .

$- \log_{3} (6)$

Step 2.3

Replace the variable $x$ with $11$ in the expression.

$f (11) = \sqrt{(11) - 5} - \log_{3} (11 - (11))$

Step 2.4

Simplify each term.

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Step 2.4.1

Subtract $5$ from $11$ .

$f (11) = \sqrt{6} - \log_{3} (11 - (11))$

Step 2.4.2

Multiply $- 1$ by $11$ .

$f (11) = \sqrt{6} - \log_{3} (11 - 11)$

Step 2.4.3

Subtract $11$ from $11$ .

$f (11) = \sqrt{6} - \log_{3} (0)$

Step 2.4.4

The logarithm of zero is undefined.

$f (11) = Undefined$

$f (11) = \sqrt{6} - \log_{3} (0)$

Step 2.5

The logarithm of zero is undefined.

$f (11) = Undefined$

Undefined

Step 3

The end points are $(5, - \log_{3} (6)), (11, Undefined)$ .

$(5, - \log_{3} (6)), (11, Undefined)$

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 $6$ into $f (x) = \sqrt{x - 5} - \log_{3} (11 - x)$ . In this case, the point is $(6, 1 - \log_{3} (5))$ .

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Step 4.1.1

Replace the variable $x$ with $6$ in the expression.

$f (6) = \sqrt{(6) - 5} - \log_{3} (11 - (6))$

Step 4.1.2

Simplify the result.

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Step 4.1.2.1

Simplify each term.

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Step 4.1.2.1.1

Subtract $5$ from $6$ .

$f (6) = \sqrt{1} - \log_{3} (11 - (6))$

Step 4.1.2.1.2

Any root of $1$ is $1$ .

$f (6) = 1 - \log_{3} (11 - (6))$

Step 4.1.2.1.3

Multiply $- 1$ by $6$ .

$f (6) = 1 - \log_{3} (11 - 6)$

Step 4.1.2.1.4

Subtract $6$ from $11$ .

$f (6) = 1 - \log_{3} (5)$

Step 4.1.2.2

The final answer is $1 - \log_{3} (5)$ .

$y = 1 - \log_{3} (5)$

Step 4.2

Substitute the $x$ value $7$ into $f (x) = \sqrt{x - 5} - \log_{3} (11 - x)$ . In this case, the point is $(7, \sqrt{2} - \log_{3} (4))$ .

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Step 4.2.1

Replace the variable $x$ with $7$ in the expression.

$f (7) = \sqrt{(7) - 5} - \log_{3} (11 - (7))$

Step 4.2.2

Simplify the result.

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Step 4.2.2.1

Simplify each term.

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Step 4.2.2.1.1

Subtract $5$ from $7$ .

$f (7) = \sqrt{2} - \log_{3} (11 - (7))$

Step 4.2.2.1.2

Multiply $- 1$ by $7$ .

$f (7) = \sqrt{2} - \log_{3} (11 - 7)$

Step 4.2.2.1.3

Subtract $7$ from $11$ .

$f (7) = \sqrt{2} - \log_{3} (4)$

Step 4.2.2.2

The final answer is $\sqrt{2} - \log_{3} (4)$ .

$y = \sqrt{2} - \log_{3} (4)$

Step 4.3

Substitute the $x$ value $8$ into $f (x) = \sqrt{x - 5} - \log_{3} (11 - x)$ . In this case, the point is $(8, \sqrt{3} - 1)$ .

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Step 4.3.1

Replace the variable $x$ with $8$ in the expression.

$f (8) = \sqrt{(8) - 5} - \log_{3} (11 - (8))$

Step 4.3.2

Simplify the result.

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Step 4.3.2.1

Simplify each term.

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Step 4.3.2.1.1

Subtract $5$ from $8$ .

$f (8) = \sqrt{3} - \log_{3} (11 - (8))$

Step 4.3.2.1.2

Multiply $- 1$ by $8$ .

$f (8) = \sqrt{3} - \log_{3} (11 - 8)$

Step 4.3.2.1.3

Subtract $8$ from $11$ .

$f (8) = \sqrt{3} - \log_{3} (3)$

Step 4.3.2.1.4

Logarithm base $3$ of $3$ is $1$ .

$f (8) = \sqrt{3} - 1 \cdot 1$

Step 4.3.2.1.5

Multiply $- 1$ by $1$ .

$f (8) = \sqrt{3} - 1$

Step 4.3.2.2

The final answer is $\sqrt{3} - 1$ .

$y = \sqrt{3} - 1$

Step 4.4

Substitute the $x$ value $9$ into $f (x) = \sqrt{x - 5} - \log_{3} (11 - x)$ . In this case, the point is $(9, 2 - \log_{3} (2))$ .

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Step 4.4.1

Replace the variable $x$ with $9$ in the expression.

$f (9) = \sqrt{(9) - 5} - \log_{3} (11 - (9))$

Step 4.4.2

Simplify the result.

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Step 4.4.2.1

Simplify each term.

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Step 4.4.2.1.1

Subtract $5$ from $9$ .

$f (9) = \sqrt{4} - \log_{3} (11 - (9))$

Step 4.4.2.1.2

Rewrite $4$ as $2^{2}$ .

$f (9) = \sqrt{2^{2}} - \log_{3} (11 - (9))$

Step 4.4.2.1.3

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

$f (9) = 2 - \log_{3} (11 - (9))$

Step 4.4.2.1.4

Multiply $- 1$ by $9$ .

$f (9) = 2 - \log_{3} (11 - 9)$

Step 4.4.2.1.5

Subtract $9$ from $11$ .

$f (9) = 2 - \log_{3} (2)$

Step 4.4.2.2

The final answer is $2 - \log_{3} (2)$ .

$y = 2 - \log_{3} (2)$

Step 4.5

Substitute the $x$ value $10$ into $f (x) = \sqrt{x - 5} - \log_{3} (11 - x)$ . In this case, the point is $(10, \sqrt{5})$ .

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Step 4.5.1

Replace the variable $x$ with $10$ in the expression.

$f (10) = \sqrt{(10) - 5} - \log_{3} (11 - (10))$

Step 4.5.2

Simplify the result.

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Step 4.5.2.1

Simplify each term.

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Step 4.5.2.1.1

Subtract $5$ from $10$ .

$f (10) = \sqrt{5} - \log_{3} (11 - (10))$

Step 4.5.2.1.2

Multiply $- 1$ by $10$ .

$f (10) = \sqrt{5} - \log_{3} (11 - 10)$

Step 4.5.2.1.3

Subtract $10$ from $11$ .

$f (10) = \sqrt{5} - \log_{3} (1)$

Step 4.5.2.1.4

Logarithm base $3$ of $1$ is $0$ .

$f (10) = \sqrt{5} - 0$

Step 4.5.2.1.5

Multiply $- 1$ by $0$ .

$f (10) = \sqrt{5} + 0$

Step 4.5.2.2

Add $\sqrt{5}$ and $0$ .

$f (10) = \sqrt{5}$

Step 4.5.2.3

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

$y = \sqrt{5}$

Step 4.6

The square root can be graphed using the points around the vertex $(5, - \log_{3} (6)), (11, Undefined), (6, - 0.46), (7, 0.15), (8, 0.73), (9, 1.37), (10, 2.24)$

$\begin{array}{c} x & y \\ 5 & - \log_{3} (6) \\ 6 & - 0.46 \\ 7 & 0.15 \\ 8 & 0.73 \\ 9 & 1.37 \\ 10 & 2.24 \\ 11 & Undefined \end{array}$

Step 5