Find the Domain and Range y=(2x)/((x-2)(x+1))

$y = \frac{2 x}{(x - 2) (x + 1)}$

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

$(x - 2) (x + 1) = 0$

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 $0$ .

$x - 2 = 0$

$x + 1 = 0$

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 $x + 1$ equal to $0$ and solve for $x$ .

Set $x + 1$ equal to $0$ .

$x + 1 = 0$

Subtract $1$ from both sides of the equation.

$x = - 1$

The final solution is all the values that make $(x - 2) (x + 1) = 0$ true.

$x = 2, - 1$

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

Interval Notation:

$(- \infty, - 1) \cup (- 1, 2) \cup (2, \infty)$

Set-Builder Notation:

${x | x \neq - 1, 2}$

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

Interval Notation:

$(- \infty, \infty)$

Set-Builder Notation:

${y | y \in ℝ}$

Determine the domain and range.

Domain: $(- \infty, - 1) \cup (- 1, 2) \cup (2, \infty), {x | x \neq - 1, 2}$

Range: $(- \infty, \infty), {y | y \in ℝ}$