Finite Math Examples

$f (x) = 18 - 3 x$ , $g (x) = 6 - \frac{x}{3}$

Step 1

Find the quotient of the functions.

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Step 1.1

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

$\frac{18 - 3 x}{6 - \frac{x}{3}}$

Step 1.2

Simplify.

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Step 1.2.1

Multiply the numerator and denominator of the fraction by $3$ .

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Step 1.2.1.1

Multiply $\frac{18 - 3 x}{6 - \frac{x}{3}}$ by $\frac{3}{3}$ .

$\frac{3}{3} \cdot \frac{18 - 3 x}{6 - \frac{x}{3}}$

Step 1.2.1.2

Combine.

$\frac{3 (18 - 3 x)}{3 (6 - \frac{x}{3})}$

Step 1.2.2

Apply the distributive property.

$\frac{3 \cdot 18 + 3 (- 3 x)}{3 \cdot 6 + 3 (- \frac{x}{3})}$

Step 1.2.3

Cancel the common factor of $3$ .

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Step 1.2.3.1

Move the leading negative in $- \frac{x}{3}$ into the numerator.

$\frac{3 \cdot 18 + 3 (- 3 x)}{3 \cdot 6 + 3 \frac{- x}{3}}$

Step 1.2.3.2

Cancel the common factor.

$\frac{3 \cdot 18 + 3 (- 3 x)}{3 \cdot 6 + 3 \frac{- x}{3}}$

Step 1.2.3.3

Rewrite the expression.

$\frac{3 \cdot 18 + 3 (- 3 x)}{3 \cdot 6 - x}$

Step 1.2.4

Simplify the numerator.

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Step 1.2.4.1

Multiply $3$ by $18$ .

$\frac{54 + 3 (- 3 x)}{3 \cdot 6 - x}$

Step 1.2.4.2

Multiply $- 3$ by $3$ .

$\frac{54 - 9 x}{3 \cdot 6 - x}$

Step 1.2.4.3

Factor $9$ out of $54 - 9 x$ .

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Step 1.2.4.3.1

Factor $9$ out of $54$ .

$\frac{9 (6) - 9 x}{3 \cdot 6 - x}$

Step 1.2.4.3.2

Factor $9$ out of $- 9 x$ .

$\frac{9 (6) + 9 (- x)}{3 \cdot 6 - x}$

Step 1.2.4.3.3

Factor $9$ out of $9 (6) + 9 (- x)$ .

$\frac{9 (6 - x)}{3 \cdot 6 - x}$

Step 1.2.5

Multiply $3$ by $6$ .

$\frac{9 (6 - x)}{18 - x}$

Step 2

Set the denominator in $\frac{9 (6 - x)}{18 - x}$ equal to $0$ to find where the expression is undefined.

$18 - x = 0$

Step 3

Solve for $x$ .

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Step 3.1

Subtract $18$ from both sides of the equation.

$- x = - 18$

Step 3.2

Divide each term in $- x = - 18$ by $- 1$ and simplify.

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Step 3.2.1

Divide each term in $- x = - 18$ by $- 1$ .

$\frac{- x}{- 1} = \frac{- 18}{- 1}$

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.

$\frac{x}{1} = \frac{- 18}{- 1}$

Step 3.2.2.2

Divide $x$ by $1$ .

$x = \frac{- 18}{- 1}$

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

Divide $- 18$ by $- 1$ .

$x = 18$

Step 4

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

Interval Notation:

$(- \infty, 18) \cup (18, \infty)$

Set-Builder Notation:

${x | x \neq 18}$

Step 5