Trigonometry Examples

$f (x) = \log (x - 2) + 2$

Step 1

Find the asymptotes.

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Step 1.1

Find where the expression $\log (x - 2) + 2$ is undefined.

$x \leq 2$

Step 1.2

Since $\log (x - 2) + 2$ $\to$ $\infty$ as $x$ $\to$ $2$ from the left and $\log (x - 2) + 2$ $\to$ $- \infty$ as $x$ $\to$ $2$ from the right, then $x = 2$ is a vertical asymptote.

$x = 2$

Step 1.3

Ignoring the logarithm, consider the rational function $R (x) = \frac{a x^{n}}{b x^{m}}$ where $n$ is the degree of the numerator and $m$ is the degree of the denominator.

1. If $n < m$ , then the x-axis, $y = 0$ , is the horizontal asymptote.

2. If $n = m$ , then the horizontal asymptote is the line $y = \frac{a}{b}$ .

3. If $n > m$ , then there is no horizontal asymptote (there is an oblique asymptote).

Step 1.4

There are no horizontal asymptotes because $Q (x)$ is $1$ .

No Horizontal Asymptotes

Step 1.5

No oblique asymptotes are present for logarithmic and trigonometric functions.

No Oblique Asymptotes

Step 1.6

This is the set of all asymptotes.

Vertical Asymptotes: $x = 2$

No Horizontal Asymptotes

Vertical Asymptotes: $x = 2$

No Horizontal Asymptotes

Step 2

Find the point at $x = 3$ .

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Step 2.1

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

$f (3) = \log ((3) - 2) + 2$

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 $2$ from $3$ .

$f (3) = \log (1) + 2$

Step 2.2.1.2

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

$f (3) = 0 + 2$

Step 2.2.2

Add $0$ and $2$ .

$f (3) = 2$

Step 2.2.3

The final answer is $2$ .

$2$

Step 2.3

Convert $2$ to decimal.

$y = 2$

Step 3

Find the point at $x = 12$ .

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Step 3.1

Replace the variable $x$ with $12$ in the expression.

$f (12) = \log ((12) - 2) + 2$

Step 3.2

Simplify the result.

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Step 3.2.1

Simplify each term.

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Step 3.2.1.1

Subtract $2$ from $12$ .

$f (12) = \log (10) + 2$

Step 3.2.1.2

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

$f (12) = 1 + 2$

Step 3.2.2

Add $1$ and $2$ .

$f (12) = 3$

Step 3.2.3

The final answer is $3$ .

$3$

Step 3.3

Convert $3$ to decimal.

$y = 3$

Step 4

Find the point at $x = 4$ .

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Step 4.1

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

$f (4) = \log ((4) - 2) + 2$

Step 4.2

Simplify the result.

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Step 4.2.1

Subtract $2$ from $4$ .

$f (4) = \log (2) + 2$

Step 4.2.2

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

$\log (2) + 2$

Step 4.3

Convert $\log (2) + 2$ to decimal.

$y = 2.30102999$

Step 5

The log function can be graphed using the vertical asymptote at $x = 2$ and the points $(3, 2), (12, 3), (4, 2.30102999)$ .

Vertical Asymptote: $x = 2$

$\begin{array}{c} x & y \\ 3 & 2 \\ 4 & 2.301 \\ 12 & 3 \end{array}$

Step 6