Trigonometry Examples

$h (x) = \log (x^{2} - 1)$

Step 1

Find the asymptotes.

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Step 1.1

Set the argument of the logarithm equal to zero.

$x^{2} - 1 = 0$

Step 1.2

Solve for $x$ .

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Step 1.2.1

Add $1$ to both sides of the equation.

$x^{2} = 1$

Step 1.2.2

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{1}$

Step 1.2.3

Any root of $1$ is $1$ .

$x = \pm 1$

Step 1.2.4

The complete solution is the result of both the positive and negative portions of the solution.

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Step 1.2.4.1

First, use the positive value of the $\pm$ to find the first solution.

$x = 1$

Step 1.2.4.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - 1$

Step 1.2.4.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = 1, - 1$

Step 1.3

The vertical asymptote occurs at $x = 1, x = - 1$ .

Vertical Asymptote: $x = 1, x = - 1$

Step 2

Find the point at $x = - 2$ .

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Step 2.1

Replace the variable $x$ with $- 2$ in the expression.

$f (- 2) = \log ({(- 2)}^{2} - 1)$

Step 2.2

Simplify the result.

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Step 2.2.1

Raise $- 2$ to the power of $2$ .

$f (- 2) = \log (4 - 1)$

Step 2.2.2

Subtract $1$ from $4$ .

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

Step 2.2.3

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

$\log (3)$

Step 2.3

Convert $\log (3)$ to decimal.

$= 0.47712125$

Step 3

Find the point at $x = - 3$ .

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Step 3.1

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

$f (- 3) = \log ({(- 3)}^{2} - 1)$

Step 3.2

Simplify the result.

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Step 3.2.1

Raise $- 3$ to the power of $2$ .

$f (- 3) = \log (9 - 1)$

Step 3.2.2

Subtract $1$ from $9$ .

$f (- 3) = \log (8)$

Step 3.2.3

The final answer is $\log (8)$ .

$\log (8)$

Step 3.3

Convert $\log (8)$ to decimal.

$= 0.90308998$

Step 4

Find the point at $x = - 4$ .

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Step 4.1

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

$f (- 4) = \log ({(- 4)}^{2} - 1)$

Step 4.2

Simplify the result.

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Step 4.2.1

Raise $- 4$ to the power of $2$ .

$f (- 4) = \log (16 - 1)$

Step 4.2.2

Subtract $1$ from $16$ .

$f (- 4) = \log (15)$

Step 4.2.3

The final answer is $\log (15)$ .

$\log (15)$

Step 4.3

Convert $\log (15)$ to decimal.

$= 1.17609125$

Step 5

The log function can be graphed using the vertical asymptote at $x = 1, x = - 1$ and the points $(- 2, 0.47712125), (- 3, 0.90308998), (- 4, 1.17609125)$ .

Vertical Asymptote: $x = 1, x = - 1$

$\begin{array}{c} x & y \\ - 4 & 1.176 \\ - 3 & 0.903 \\ - 2 & 0.477 \end{array}$

Step 6