Find the Domain f(x) = natural log of x^2+5x-6

$f (x) = \ln (x^{2} + 5 x - 6)$

Step 1

Set the argument in $\ln (x^{2} + 5 x - 6)$ greater than $0$ to find where the expression is defined.

$x^{2} + 5 x - 6 > 0$

Step 2

Solve for $x$ .

Convert the inequality to an equation.

$x^{2} + 5 x - 6 = 0$

Factor $x^{2} + 5 x - 6$ 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 $- 6$ and whose sum is $5$ .

$- 1, 6$

Write the factored form using these integers.

$(x - 1) (x + 6) = 0$

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

$x - 1 = 0$

$x + 6 = 0$

Set $x - 1$ equal to $0$ and solve for $x$ .

Set $x - 1$ equal to $0$ .

$x - 1 = 0$

Add $1$ to both sides of the equation.

$x = 1$

Set $x + 6$ equal to $0$ and solve for $x$ .

Set $x + 6$ equal to $0$ .

$x + 6 = 0$

Subtract $6$ from both sides of the equation.

$x = - 6$

The final solution is all the values that make $(x - 1) (x + 6) = 0$ true.

$x = 1, - 6$

Use each root to create test intervals.

$x < - 6$

$- 6 < x < 1$

$x > 1$

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 $x < - 6$ to see if it makes the inequality true.

Choose a value on the interval $x < - 6$ and see if this value makes the original inequality true.

$x = - 8$

Replace $x$ with $- 8$ in the original inequality.

${(- 8)}^{2} + 5 (- 8) - 6 > 0$

The left side $18$ is greater than the right side $0$ , which means that the given statement is always true.

True

Test a value on the interval $- 6 < x < 1$ to see if it makes the inequality true.

Choose a value on the interval $- 6 < x < 1$ and see if this value makes the original inequality true.

$x = 0$

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

${(0)}^{2} + 5 (0) - 6 > 0$

The left side $- 6$ is not greater than the right side $0$ , which means that the given statement is false.

False

Test a value on the interval $x > 1$ to see if it makes the inequality true.

Choose a value on the interval $x > 1$ and see if this value makes the original inequality true.

$x = 4$

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

${(4)}^{2} + 5 (4) - 6 > 0$

The left side $30$ 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.

$x < - 6$ True

$- 6 < x < 1$ False

$x > 1$ True

$x < - 6$ True

$- 6 < x < 1$ False

$x > 1$ True

The solution consists of all of the true intervals.

$x < - 6$ or $x > 1$

Step 3

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

Interval Notation:

$(- \infty, - 6) \cup (1, \infty)$

Set-Builder Notation:

${x | x < - 6, x > 1}$

Step 4

Find the Domain f(x) = natural log of x^2+5x-6

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