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

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

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

$x (x - 3) = 0$

Solve for $x$ .

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x = 0$

$x - 3 = 0$

Set $x$ equal to $0$ .

$x = 0$

Set $x - 3$ equal to $0$ and solve for $x$ .

Set $x - 3$ equal to $0$ .

$x - 3 = 0$

Add $3$ to both sides of the equation.

$x = 3$

The final solution is all the values that make $x (x - 3) = 0$ true.

$x = 0, 3$

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

Interval Notation:

$(- \infty, 0) \cup (0, 3) \cup (3, \infty)$

Set-Builder Notation:

${x | x \neq 0, 3}$

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

Interval Notation:

$(- \infty, 0) \cup (0, 3) \cup (3, \infty)$

Set-Builder Notation:

${y | y \neq 0, 3}$

Determine the domain and range.

Domain: $(- \infty, 0) \cup (0, 3) \cup (3, \infty), {x | x \neq 0, 3}$

Range: $(- \infty, 0) \cup (0, 3) \cup (3, \infty), {y | y \neq 0, 3}$