Algebra Examples

$\frac{1}{| x - 2 |}$

Step 1

Set the denominator in $\frac{1}{| x - 2 |}$ equal to $0$ to find where the expression is undefined.

$| x - 2 | = 0$

Step 2

Solve for $x$ .

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Step 2.1

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

$x - 2 = \pm 0$

Step 2.2

Plus or minus $0$ is $0$ .

$x - 2 = 0$

Step 2.3

Add $2$ to both sides of the equation.

$x = 2$

Step 3

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

Interval Notation:

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

Set-Builder Notation:

${x | x \neq 2}$

Step 4

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

Interval Notation:

$(0, \infty)$

Set-Builder Notation:

${y | y > 0}$

Step 5

Determine the domain and range.

Domain: $(- \infty, 2) \cup (2, \infty), {x | x \neq 2}$

Range: $(0, \infty), {y | y > 0}$

Step 6