Find the Domain of the Quotient of the Two Functions f(x)=6/(1-7x) , g(x)=1/x

$f (x) = \frac{6}{1 - 7 x}$ , $g (x) = \frac{1}{x}$

Step 1

Find the quotient of the functions.

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

$\frac{\frac{6}{1 - 7 x}}{\frac{1}{x}}$

Simplify.

Multiply the numerator by the reciprocal of the denominator.

$\frac{6}{1 - 7 x} x$

Combine $\frac{6}{1 - 7 x}$ and $x$ .

$\frac{6 x}{1 - 7 x}$

Step 2

Set the denominator in $\frac{6 x}{1 - 7 x}$ equal to $0$ to find where the expression is undefined.

$1 - 7 x = 0$

Step 3

Solve for $x$ .

Subtract $1$ from both sides of the equation.

$- 7 x = - 1$

Divide each term in $- 7 x = - 1$ by $- 7$ and simplify.

Divide each term in $- 7 x = - 1$ by $- 7$ .

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

Simplify the left side.

Cancel the common factor of $- 7$ .

Cancel the common factor.

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

Divide $x$ by $1$ .

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

Simplify the right side.

Dividing two negative values results in a positive value.

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

Step 4

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

Interval Notation:

$(- \infty, \frac{1}{7}) \cup (\frac{1}{7}, \infty)$

Set-Builder Notation:

${x | x \neq \frac{1}{7}}$

Step 5