Find the Domain and Range (3x+6)/(-6x+2)

$\frac{3 x + 6}{- 6 x + 2}$

Step 1

Set the denominator in $\frac{3 x + 6}{- 6 x + 2}$ equal to $0$ to find where the expression is undefined.

$- 6 x + 2 = 0$

Step 2

Solve for $x$ .

Subtract $2$ from both sides of the equation.

$- 6 x = - 2$

Divide each term in $- 6 x = - 2$ by $- 6$ and simplify.

Divide each term in $- 6 x = - 2$ by $- 6$ .

$\frac{- 6 x}{- 6} = \frac{- 2}{- 6}$

Simplify the left side.

Cancel the common factor of $- 6$ .

Cancel the common factor.

$\frac{- 6 x}{- 6} = \frac{- 2}{- 6}$

Divide $x$ by $1$ .

$x = \frac{- 2}{- 6}$

Simplify the right side.

Cancel the common factor of $- 2$ and $- 6$ .

Factor $- 2$ out of $- 2$ .

$x = \frac{- 2 (1)}{- 6}$

Cancel the common factors.

Factor $- 2$ out of $- 6$ .

$x = \frac{- 2 \cdot 1}{- 2 \cdot 3}$

Cancel the common factor.

$x = \frac{- 2 \cdot 1}{- 2 \cdot 3}$

Rewrite the expression.

$x = \frac{1}{3}$

Step 3

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

Interval Notation:

$(- \infty, \frac{1}{3}) \cup (\frac{1}{3}, \infty)$

Set-Builder Notation:

${x | x \neq \frac{1}{3}}$

Step 4

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

Interval Notation:

$(- \infty, - \frac{1}{2}) \cup (- \frac{1}{2}, \infty)$

Set-Builder Notation:

${y | y \neq - \frac{1}{2}}$

Step 5

Determine the domain and range.

Domain: $(- \infty, \frac{1}{3}) \cup (\frac{1}{3}, \infty), {x | x \neq \frac{1}{3}}$

Range: $(- \infty, - \frac{1}{2}) \cup (- \frac{1}{2}, \infty), {y | y \neq - \frac{1}{2}}$

Step 6