Precalculus Examples

$\frac{y^{2} + 1}{y^{2} - 2 y}$

Step 1

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

$y^{2} - 2 y = 0$

Step 2

Solve for $y$ .

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Step 2.1

Factor $y$ out of $y^{2} - 2 y$ .

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Step 2.1.1

Factor $y$ out of $y^{2}$ .

$y \cdot y - 2 y = 0$

Step 2.1.2

Factor $y$ out of $- 2 y$ .

$y \cdot y + y \cdot - 2 = 0$

Step 2.1.3

Factor $y$ out of $y \cdot y + y \cdot - 2$ .

$y (y - 2) = 0$

Step 2.2

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$y = 0$

$y - 2 = 0$

Step 2.3

Set $y$ equal to $0$ .

$y = 0$

Step 2.4

Set $y - 2$ equal to $0$ and solve for $y$ .

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Step 2.4.1

Set $y - 2$ equal to $0$ .

$y - 2 = 0$

Step 2.4.2

Add $2$ to both sides of the equation.

$y = 2$

Step 2.5

The final solution is all the values that make $y (y - 2) = 0$ true.

$y = 0, 2$

Step 3

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

Interval Notation:

$(- \infty, 0) \cup (0, 2) \cup (2, \infty)$

Set-Builder Notation:

${y | y \neq 0, 2}$

Step 4