Find the Domain and Range (x+3)/(x-5)

$\frac{x + 3}{x - 5}$

Step 1

Set the denominator in $\frac{x + 3}{x - 5}$ equal to $0$ to find where the expression is undefined.

$x - 5 = 0$

Step 2

Add $5$ to both sides of the equation.

$x = 5$

Step 3

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

Interval Notation:

$(- \infty, 5) \cup (5, \infty)$

Set-Builder Notation:

${x | x \neq 5}$

Step 4

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

Interval Notation:

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

Set-Builder Notation:

${y | y \neq 1}$

Step 5

Determine the domain and range.

Domain: $(- \infty, 5) \cup (5, \infty), {x | x \neq 5}$

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

Step 6