Find the Domain and Range f(x)=1/(|x|-3)

$f (x) = \frac{1}{| x | - 3}$

Step 1

Set the denominator in $\frac{1}{| x | - 3}$ equal to $0$ to find where the expression is undefined.

$| x | - 3 = 0$

Step 2

Solve for $x$ .

Add $3$ to both sides of the equation.

$| x | = 3$

Remove the absolute value term. This creates a $\pm$ on the right side of the equation because $| x | = \pm x$ .

$x = \pm 3$

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

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

$x = 3$

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

$x = - 3$

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

$x = 3, - 3$

Step 3

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

Interval Notation:

$(- \infty, - 3) \cup (- 3, 3) \cup (3, \infty)$

Set-Builder Notation:

${x | x \neq 3, - 3}$

Step 4

The range is the set of all valid $y$ values. Use the graph to find the range.

Interval Notation:

$(- \infty, - \frac{1}{3}] \cup (0, \infty)$

Set-Builder Notation:

${y | y \leq - \frac{1}{3}, y > 0}$

Step 5

Determine the domain and range.

Domain: $(- \infty, - 3) \cup (- 3, 3) \cup (3, \infty), {x | x \neq 3, - 3}$

Range: $(- \infty, - \frac{1}{3}] \cup (0, \infty), {y | y \leq - \frac{1}{3}, y > 0}$

Step 6