Finite Math Examples

$f (x) = \frac{x}{x - 7}$ , $g (x) = - \frac{1}{x}$

Step 1

Find the product of the functions.

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Step 1.1

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

$(\frac{x}{x - 7}) \cdot (- \frac{1}{x})$

Step 1.2

Simplify.

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Step 1.2.1

Rewrite using the commutative property of multiplication.

$- \frac{x}{x - 7} \cdot \frac{1}{x}$

Step 1.2.2

Cancel the common factor of $x$ .

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Step 1.2.2.1

Move the leading negative in $- \frac{x}{x - 7}$ into the numerator.

$\frac{- x}{x - 7} \cdot \frac{1}{x}$

Step 1.2.2.2

Factor $x$ out of $- x$ .

$\frac{x \cdot - 1}{x - 7} \cdot \frac{1}{x}$

Step 1.2.2.3

Cancel the common factor.

$\frac{x \cdot - 1}{x - 7} \cdot \frac{1}{x}$

Step 1.2.2.4

Rewrite the expression.

$\frac{- 1}{x - 7}$

Step 1.2.3

Move the negative in front of the fraction.

$- \frac{1}{x - 7}$

Step 2

Set the denominator in $\frac{1}{x - 7}$ equal to $0$ to find where the expression is undefined.

$x - 7 = 0$

Step 3

Add $7$ to both sides of the equation.

$x = 7$

Step 4

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

Interval Notation:

$(- \infty, 7) \cup (7, \infty)$

Set-Builder Notation:

${x | x \neq 7}$

Step 5