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

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

Step 1

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

$(x - 2) (x + 2) = 0$

Step 2

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 $0$ .

$x - 2 = 0$

$x + 2 = 0$

Set $x - 2$ equal to $0$ and solve for $x$ .

Set $x - 2$ equal to $0$ .

$x - 2 = 0$

Add $2$ to both sides of the equation.

$x = 2$

Set $x + 2$ equal to $0$ and solve for $x$ .

Set $x + 2$ equal to $0$ .

$x + 2 = 0$

Subtract $2$ from both sides of the equation.

$x = - 2$

The final solution is all the values that make $(x - 2) (x + 2) = 0$ true.

$x = 2, - 2$

Step 3

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

Interval Notation:

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

Set-Builder Notation:

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

Step 4

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

Interval Notation:

$(- \infty, - 3) \cup (- 3, - \frac{3}{2}) \cup (- \frac{3}{2}, \infty)$

Set-Builder Notation:

${y | y \neq - \frac{3}{2}, - 3}$

Step 5

Determine the domain and range.

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

Range: $(- \infty, - 3) \cup (- 3, - \frac{3}{2}) \cup (- \frac{3}{2}, \infty), {y | y \neq - \frac{3}{2}, - 3}$

Step 6