Finite Math Examples

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

Step 1

Find the sum of the functions.

Tap for more steps...

Step 1.1

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

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

Step 1.2

Simplify.

Tap for more steps...

Step 1.2.1

To write $\frac{x}{x - 7}$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

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

Step 1.2.2

To write $- \frac{1}{x}$ as a fraction with a common denominator, multiply by $\frac{x - 7}{x - 7}$ .

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

Step 1.2.3

Write each expression with a common denominator of $(x - 7) x$ , by multiplying each by an appropriate factor of $1$ .

Tap for more steps...

Step 1.2.3.1

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

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

Step 1.2.3.2

Multiply $\frac{1}{x}$ by $\frac{x - 7}{x - 7}$ .

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

Step 1.2.3.3

Reorder the factors of $(x - 7) x$ .

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

Step 1.2.4

Combine the numerators over the common denominator.

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

Step 1.2.5

Simplify the numerator.

Tap for more steps...

Step 1.2.5.1

Multiply $x$ by $x$ .

$\frac{x^{2} - (x - 7)}{x (x - 7)}$

Step 1.2.5.2

Apply the distributive property.

$\frac{x^{2} - x - - 7}{x (x - 7)}$

Step 1.2.5.3

Multiply $- 1$ by $- 7$ .

$\frac{x^{2} - x + 7}{x (x - 7)}$

Step 2

Set the denominator in $\frac{x^{2} - x + 7}{x (x - 7)}$ equal to $0$ to find where the expression is undefined.

$x (x - 7) = 0$

Step 3

Solve for $x$ .

Tap for more steps...

Step 3.1

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x = 0$

$x - 7 = 0$

Step 3.2

Set $x$ equal to $0$ .

$x = 0$

Step 3.3

Set $x - 7$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 3.3.1

Set $x - 7$ equal to $0$ .

$x - 7 = 0$

Step 3.3.2

Add $7$ to both sides of the equation.

$x = 7$

Step 3.4

The final solution is all the values that make $x (x - 7) = 0$ true.

$x = 0, 7$

Step 4

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

Interval Notation:

$(- \infty, 0) \cup (0, 7) \cup (7, \infty)$

Set-Builder Notation:

${x | x \neq 0, 7}$

Step 5