Finite Math Examples

$f (x) = \frac{7}{x - 8}$ , $g (x) = \frac{5}{x}$

Step 1

Find the quotient of the functions.

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Step 1.1

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

$\frac{\frac{7}{x - 8}}{\frac{5}{x}}$

Step 1.2

Simplify.

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Step 1.2.1

Multiply the numerator by the reciprocal of the denominator.

$\frac{7}{x - 8} \cdot \frac{x}{5}$

Step 1.2.2

Multiply $\frac{7}{x - 8}$ by $\frac{x}{5}$ .

$\frac{7 x}{(x - 8) \cdot 5}$

Step 1.2.3

Move $5$ to the left of $x - 8$ .

$\frac{7 x}{5 (x - 8)}$

Step 2

Set the denominator in $\frac{7 x}{5 (x - 8)}$ equal to $0$ to find where the expression is undefined.

$5 (x - 8) = 0$

Step 3

Solve for $x$ .

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Step 3.1

Divide each term in $5 (x - 8) = 0$ by $5$ and simplify.

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Step 3.1.1

Divide each term in $5 (x - 8) = 0$ by $5$ .

$\frac{5 (x - 8)}{5} = \frac{0}{5}$

Step 3.1.2

Simplify the left side.

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Step 3.1.2.1

Cancel the common factor of $5$ .

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Step 3.1.2.1.1

Cancel the common factor.

$\frac{5 (x - 8)}{5} = \frac{0}{5}$

Step 3.1.2.1.2

Divide $x - 8$ by $1$ .

$x - 8 = \frac{0}{5}$

Step 3.1.3

Simplify the right side.

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Step 3.1.3.1

Divide $0$ by $5$ .

$x - 8 = 0$

Step 3.2

Add $8$ to both sides of the equation.

$x = 8$

Step 4

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

Interval Notation:

$(- \infty, 8) \cup (8, \infty)$

Set-Builder Notation:

${x | x \neq 8}$

Step 5