Precalculus Examples

$\frac{x - y}{x} - \frac{y - x}{y} + \frac{x^{2} + y^{2}}{x y}$

Step 1

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

$x = 0$

Step 2

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

$y = 0$

Step 3

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 4

Divide each term in $x y = 0$ by $y$ and simplify.

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Step 4.1

Divide each term in $x y = 0$ by $y$ .

$\frac{x y}{y} = \frac{0}{y}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $y$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{x y}{y} = \frac{0}{y}$

Step 4.2.1.2

Divide $x$ by $1$ .

$x = \frac{0}{y}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Divide $0$ by $y$ .

$x = 0$

Step 5

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

Interval Notation:

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

Set-Builder Notation:

${x | x \neq 0}$