Precalculus Examples

$f (x) = 5 (\frac{1}{x - 4} - \frac{1}{x + 2})$

Step 1

Set the denominator in $\frac{1}{x - 4}$ equal to $0$ to find where the expression is undefined.

$x - 4 = 0$

Step 2

Add $4$ to both sides of the equation.

$x = 4$

Step 3

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

$x + 2 = 0$

Step 4

Subtract $2$ from both sides of the equation.

$x = - 2$

Step 5

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

Interval Notation:

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

Set-Builder Notation:

${x | x \neq - 2, 4}$

Step 6

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

Interval Notation:

$(- \infty, - \frac{10}{3}] \cup (0, \infty)$

Set-Builder Notation:

${y | y \leq - \frac{10}{3}, y > 0}$

Step 7

Determine the domain and range.

Domain: $(- \infty, - 2) \cup (- 2, 4) \cup (4, \infty), {x | x \neq - 2, 4}$

Range: $(- \infty, - \frac{10}{3}] \cup (0, \infty), {y | y \leq - \frac{10}{3}, y > 0}$

Step 8