Calculus Examples

$f (x) = \frac{x^{2 - 3}}{x - 2}$

Step 1

Set the denominator in $\frac{x^{2 - 3}}{x - 2}$ equal to $0$ to find where the expression is undefined.

$x - 2 = 0$

Step 2

Add $2$ to both sides of the equation.

$x = 2$

Step 3

Set the base in $x^{2 - 3}$ equal to $0$ to find where the expression is undefined.

$x = 0$

Step 4

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

Interval Notation:

$(- \infty, 0) \cup (0, 2) \cup (2, \infty)$

Set-Builder Notation:

${x | x \neq 0, 2}$

Step 5

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

Interval Notation:

$(- \infty, - 1] \cup (0, \infty)$

Set-Builder Notation:

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

Step 6

Determine the domain and range.

Domain: $(- \infty, 0) \cup (0, 2) \cup (2, \infty), {x | x \neq 0, 2}$

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

Step 7