Precalculus Examples

$f (x) = \ln (\frac{1}{x - 9})$

Step 1

Set the argument in $\ln (\frac{1}{x - 9})$ greater than $0$ to find where the expression is defined.

$\frac{1}{x - 9} > 0$

Step 2

Solve for $x$ .

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Step 2.1

Find all the values where the expression switches from negative to positive by setting each factor equal to $0$ and solving.

$x - 9 = 0$

Step 2.2

Add $9$ to both sides of the equation.

$x = 9$

Step 2.3

Find the domain of $\frac{1}{x - 9}$ .

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Step 2.3.1

Set the denominator in $\frac{1}{x - 9}$ equal to $0$ to find where the expression is undefined.

$x - 9 = 0$

Step 2.3.2

Add $9$ to both sides of the equation.

$x = 9$

Step 2.3.3

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

$(- \infty, 9) \cup (9, \infty)$

Step 2.4

The solution consists of all of the true intervals.

$x > 9$

Step 3

Set the denominator in $\frac{1}{x - 9}$ equal to $0$ to find where the expression is undefined.

$x - 9 = 0$

Step 4

Add $9$ to both sides of the equation.

$x = 9$

Step 5

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

Interval Notation:

$(9, \infty)$

Set-Builder Notation:

${x | x > 9}$

Step 6