Algebra Examples

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

Step 1

Find the quotient of the functions.

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Step 1.1

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

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

Step 1.2

Simplify.

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Step 1.2.1

Multiply the numerator by the reciprocal of the denominator.

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

Step 1.2.2

Cancel the common factor of $x - 1$ .

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Step 1.2.2.1

Cancel the common factor.

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

Step 1.2.2.2

Rewrite the expression.

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

Step 1.2.3

Cancel the common factor of $x$ .

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Step 1.2.3.1

Factor $x$ out of $2 x$ .

$\frac{1}{x \cdot 2} x$

Step 1.2.3.2

Cancel the common factor.

$\frac{1}{x \cdot 2} x$

Step 1.2.3.3

Rewrite the expression.

$\frac{1}{2}$

Step 2

The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.

Interval Notation:

$(- \infty, \infty)$

Set-Builder Notation:

${x | x \in ℝ}$

Step 3