Find the Domain f(x) = natural log of x^2-2x-8

$f (x) = \ln (x^{2} - 2 x - 8)$

Step 1

Set the argument in $\ln (x^{2} - 2 x - 8)$ greater than $0$ to find where the expression is defined.

$x^{2} - 2 x - 8 > 0$

Step 2

Solve for $x$ .

Convert the inequality to an equation.

$x^{2} - 2 x - 8 = 0$

Factor $x^{2} - 2 x - 8$ using the AC method.

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 8$ and whose sum is $- 2$ .

$- 4, 2$

Write the factored form using these integers.

$(x - 4) (x + 2) = 0$

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

$x - 4 = 0$

$x + 2 = 0$

Set $x - 4$ equal to $0$ and solve for $x$ .

Set $x - 4$ equal to $0$ .

$x - 4 = 0$

Add $4$ to both sides of the equation.

$x = 4$

Set $x + 2$ equal to $0$ and solve for $x$ .

Set $x + 2$ equal to $0$ .

$x + 2 = 0$

Subtract $2$ from both sides of the equation.

$x = - 2$

The final solution is all the values that make $(x - 4) (x + 2) = 0$ true.

$x = 4, - 2$

Use each root to create test intervals.

$x < - 2$

$- 2 < x < 4$

$x > 4$

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

Test a value on the interval $x < - 2$ to see if it makes the inequality true.

Choose a value on the interval $x < - 2$ and see if this value makes the original inequality true.

$x = - 4$

Replace $x$ with $- 4$ in the original inequality.

${(- 4)}^{2} - 2 \cdot - 4 - 8 > 0$

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

True

Test a value on the interval $- 2 < x < 4$ to see if it makes the inequality true.

Choose a value on the interval $- 2 < x < 4$ and see if this value makes the original inequality true.

$x = 0$

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

${(0)}^{2} - 2 \cdot 0 - 8 > 0$

The left side $- 8$ is not greater than the right side $0$ , which means that the given statement is false.

False

Test a value on the interval $x > 4$ to see if it makes the inequality true.

Choose a value on the interval $x > 4$ and see if this value makes the original inequality true.

$x = 6$

Replace $x$ with $6$ in the original inequality.

${(6)}^{2} - 2 \cdot 6 - 8 > 0$

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

True

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

$x < - 2$ True

$- 2 < x < 4$ False

$x > 4$ True

$x < - 2$ True

$- 2 < x < 4$ False

$x > 4$ True

The solution consists of all of the true intervals.

$x < - 2$ or $x > 4$

Step 3

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

Interval Notation:

$(- \infty, - 2) \cup (4, \infty)$

Set-Builder Notation:

${x | x < - 2, x > 4}$

Step 4

Find the Domain f(x) = natural log of x^2-2x-8

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