Precalculus Examples

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

Step 1

Set the denominator in $\frac{1}{y}$ equal to $0$ to find where the expression is undefined.

$y = 0$

Step 2

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

$x - y = 0$

Step 3

Solve for $y$ .

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Step 3.1

Subtract $x$ from both sides of the equation.

$- y = - x$

Step 3.2

Divide each term in $- y = - x$ by $- 1$ and simplify.

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Step 3.2.1

Divide each term in $- y = - x$ by $- 1$ .

$\frac{- y}{- 1} = \frac{- x}{- 1}$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Dividing two negative values results in a positive value.

$\frac{y}{1} = \frac{- x}{- 1}$

Step 3.2.2.2

Divide $y$ by $1$ .

$y = \frac{- x}{- 1}$

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

Dividing two negative values results in a positive value.

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

Step 3.2.3.2

Divide $x$ by $1$ .

$y = x$

Step 4

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

$x + y = 0$

Step 5

Subtract $x$ from both sides of the equation.

$y = - x$

Step 6

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

Interval Notation:

$(- \infty, 0) \cup (0, \infty)$

Set-Builder Notation:

${y | y \neq 0}$