Find the Domain of the Sum of the Functions f(x)=2/(x-3) g(x)=6/x

$f (x) = \frac{2}{x - 3}$ $g (x) = \frac{6}{x}$

Step 1

Find the sum of the functions.

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

$\frac{2}{x - 3} + \frac{6}{x}$

Simplify.

To write $\frac{2}{x - 3}$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

$\frac{2}{x - 3} \cdot \frac{x}{x} + \frac{6}{x}$

To write $\frac{6}{x}$ as a fraction with a common denominator, multiply by $\frac{x - 3}{x - 3}$ .

$\frac{2}{x - 3} \cdot \frac{x}{x} + \frac{6}{x} \cdot \frac{x - 3}{x - 3}$

Write each expression with a common denominator of $(x - 3) x$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{2}{x - 3}$ by $\frac{x}{x}$ .

$\frac{2 x}{(x - 3) x} + \frac{6}{x} \cdot \frac{x - 3}{x - 3}$

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

$\frac{2 x}{(x - 3) x} + \frac{6 (x - 3)}{x (x - 3)}$

Reorder the factors of $(x - 3) x$ .

$\frac{2 x}{x (x - 3)} + \frac{6 (x - 3)}{x (x - 3)}$

Combine the numerators over the common denominator.

$\frac{2 x + 6 (x - 3)}{x (x - 3)}$

Simplify the numerator.

Factor $2$ out of $2 x + 6 (x - 3)$ .

Factor $2$ out of $2 x$ .

$\frac{2 (x) + 6 (x - 3)}{x (x - 3)}$

Factor $2$ out of $6 (x - 3)$ .

$\frac{2 (x) + 2 (3 (x - 3))}{x (x - 3)}$

Factor $2$ out of $2 (x) + 2 (3 (x - 3))$ .

$\frac{2 (x + 3 (x - 3))}{x (x - 3)}$

Apply the distributive property.

$\frac{2 (x + 3 x + 3 \cdot - 3)}{x (x - 3)}$

Multiply $3$ by $- 3$ .

$\frac{2 (x + 3 x - 9)}{x (x - 3)}$

Add $x$ and $3 x$ .

$\frac{2 (4 x - 9)}{x (x - 3)}$

Step 2

Set the denominator in $\frac{2 (4 x - 9)}{x (x - 3)}$ equal to $0$ to find where the expression is undefined.

$x (x - 3) = 0$

Step 3

Solve for $x$ .

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

$x = 0$

$x - 3 = 0$

Set $x$ equal to $0$ .

$x = 0$

Set $x - 3$ equal to $0$ and solve for $x$ .

Set $x - 3$ equal to $0$ .

$x - 3 = 0$

Add $3$ to both sides of the equation.

$x = 3$

The final solution is all the values that make $x (x - 3) = 0$ true.

$x = 0, 3$

Step 4

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

Interval Notation:

$(- \infty, 0) \cup (0, 3) \cup (3, \infty)$

Set-Builder Notation:

${x | x \neq 0, 3}$

Step 5

Find the Domain of the Sum of the Functions f(x)=2/(x-3) g(x)=6/x

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