Trigonometry Examples

$f (x) = \log (12 - x)$

Step 1

Find the asymptotes.

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Step 1.1

Set the argument of the logarithm equal to zero.

$12 - x = 0$

Step 1.2

Solve for $x$ .

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Step 1.2.1

Subtract $12$ from both sides of the equation.

$- x = - 12$

Step 1.2.2

Divide each term in $- x = - 12$ by $- 1$ and simplify.

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Step 1.2.2.1

Divide each term in $- x = - 12$ by $- 1$ .

$\frac{- x}{- 1} = \frac{- 12}{- 1}$

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.

$\frac{x}{1} = \frac{- 12}{- 1}$

Step 1.2.2.2.2

Divide $x$ by $1$ .

$x = \frac{- 12}{- 1}$

Step 1.2.2.3

Simplify the right side.

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Step 1.2.2.3.1

Divide $- 12$ by $- 1$ .

$x = 12$

Step 1.3

The vertical asymptote occurs at $x = 12$ .

Vertical Asymptote: $x = 12$

Step 2

Find the point at $x = 11$ .

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Step 2.1

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

$f (11) = \log (12 - (11))$

Step 2.2

Simplify the result.

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Step 2.2.1

Multiply $- 1$ by $11$ .

$f (11) = \log (12 - 11)$

Step 2.2.2

Subtract $11$ from $12$ .

$f (11) = \log (1)$

Step 2.2.3

Logarithm base $10$ of $1$ is $0$ .

$f (11) = 0$

Step 2.2.4

The final answer is $0$ .

$0$

Step 2.3

Convert $0$ to decimal.

$= 0$

Step 3

Find the point at $x = 10$ .

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Step 3.1

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

$f (10) = \log (12 - (10))$

Step 3.2

Simplify the result.

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Step 3.2.1

Multiply $- 1$ by $10$ .

$f (10) = \log (12 - 10)$

Step 3.2.2

Subtract $10$ from $12$ .

$f (10) = \log (2)$

Step 3.2.3

The final answer is $\log (2)$ .

$\log (2)$

Step 3.3

Convert $\log (2)$ to decimal.

$= 0.30102999$

Step 4

Find the point at $x = 9$ .

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Step 4.1

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

$f (9) = \log (12 - (9))$

Step 4.2

Simplify the result.

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Step 4.2.1

Multiply $- 1$ by $9$ .

$f (9) = \log (12 - 9)$

Step 4.2.2

Subtract $9$ from $12$ .

$f (9) = \log (3)$

Step 4.2.3

The final answer is $\log (3)$ .

$\log (3)$

Step 4.3

Convert $\log (3)$ to decimal.

$= 0.47712125$

Step 5

The log function can be graphed using the vertical asymptote at $x = 12$ and the points $(11, 0), (10, 0.30102999), (9, 0.47712125)$ .

Vertical Asymptote: $x = 12$

$\begin{array}{c} x & y \\ 9 & 0.477 \\ 10 & 0.301 \\ 11 & 0 \end{array}$

Step 6