Examples

$f (x) = x^{2} - 1$ , $g (x) = 2 x$

Step 1

Find the quotient of the functions.

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Step 1.1

Replace the function designators with the actual functions in $\frac{f (x)}{g (x)}$ .

$\frac{x^{2} - 1}{2 x}$

Step 1.2

Simplify the numerator.

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Step 1.2.1

Rewrite $1$ as $1^{2}$ .

$\frac{x^{2} - 1^{2}}{2 x}$

Step 1.2.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 1$ .

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

Step 2

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

$2 x = 0$

Step 3

Divide each term in $2 x = 0$ by $2$ and simplify.

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Step 3.1

Divide each term in $2 x = 0$ by $2$ .

$\frac{2 x}{2} = \frac{0}{2}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $2$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{0}{2}$

Step 3.2.1.2

Divide $x$ by $1$ .

$x = \frac{0}{2}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Divide $0$ by $2$ .

$x = 0$

Step 4

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

Interval Notation:

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

Set-Builder Notation:

${x | x \neq 0}$

Step 5

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