Find the Domain and Range 3y+xy=2x+1

$3 y + x y = 2 x + 1$

Step 1

Factor $y$ out of $3 y + x y$ .

Factor $y$ out of $3 y$ .

$y \cdot 3 + x y = 2 x + 1$

Factor $y$ out of $x y$ .

$y \cdot 3 + y x = 2 x + 1$

Factor $y$ out of $y \cdot 3 + y x$ .

$y (3 + x) = 2 x + 1$

Step 2

Divide each term in $y (3 + x) = 2 x + 1$ by $3 + x$ and simplify.

Divide each term in $y (3 + x) = 2 x + 1$ by $3 + x$ .

$\frac{y (3 + x)}{3 + x} = \frac{2 x}{3 + x} + \frac{1}{3 + x}$

Simplify the left side.

Cancel the common factor of $3 + x$ .

Cancel the common factor.

$\frac{y (3 + x)}{3 + x} = \frac{2 x}{3 + x} + \frac{1}{3 + x}$

Divide $y$ by $1$ .

$y = \frac{2 x}{3 + x} + \frac{1}{3 + x}$

Simplify the right side.

Combine the numerators over the common denominator.

$y = \frac{2 x + 1}{3 + x}$

Step 3

Set the denominator in $\frac{2 x + 1}{3 + x}$ equal to $0$ to find where the expression is undefined.

$3 + x = 0$

Step 4

Subtract $3$ from both sides of the equation.

$x = - 3$

Step 5

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

Interval Notation:

$(- \infty, - 3) \cup (- 3, \infty)$

Set-Builder Notation:

${x | x \neq - 3}$

Step 6

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

Interval Notation:

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

Set-Builder Notation:

${y | y \neq 2}$

Step 7

Determine the domain and range.

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

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

Step 8