Find the Domain and Range y=(x+6)/(4x-8)

$y = \frac{x + 6}{4 x - 8}$

Step 1

Set the denominator in $\frac{x + 6}{4 x - 8}$ equal to $0$ to find where the expression is undefined.

$4 x - 8 = 0$

Step 2

Solve for $x$ .

Add $8$ to both sides of the equation.

$4 x = 8$

Divide each term in $4 x = 8$ by $4$ and simplify.

Divide each term in $4 x = 8$ by $4$ .

$\frac{4 x}{4} = \frac{8}{4}$

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 x}{4} = \frac{8}{4}$

Divide $x$ by $1$ .

$x = \frac{8}{4}$

Simplify the right side.

Divide $8$ by $4$ .

$x = 2$

Step 3

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

Interval Notation:

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

Set-Builder Notation:

${x | x \neq 2}$

Step 4

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

Interval Notation:

$(- \infty, \frac{1}{4}) \cup (\frac{1}{4}, \infty)$

Set-Builder Notation:

${y | y \neq \frac{1}{4}}$

Step 5

Determine the domain and range.

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

Range: $(- \infty, \frac{1}{4}) \cup (\frac{1}{4}, \infty), {y | y \neq \frac{1}{4}}$

Step 6