Find the Domain and Range r(x)=(x^3-3x^2-4x)/(x-4)

$r (x) = \frac{x^{3} - 3 x^{2} - 4 x}{x - 4}$

Set the denominator in $\frac{x^{3} - 3 x^{2} - 4 x}{x - 4}$ equal to $0$ to find where the expression is undefined.

$x - 4 = 0$

Add $4$ to both sides of the equation.

$x = 4$

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

Interval Notation:

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

Set-Builder Notation:

${x | x \neq 4}$

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

Interval Notation:

$[- \frac{1}{4}, \infty)$

Set-Builder Notation:

${y | y \geq - \frac{1}{4}}$

Determine the domain and range.

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

Range: $[- \frac{1}{4}, \infty), {y | y \geq - \frac{1}{4}}$