Algebra Examples

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

Step 1

Find the product of the functions.

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Step 1.1

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

$(\frac{6}{1 - 7 x}) \cdot (\frac{1}{x})$

Step 1.2

Simplify.

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Step 1.2.1

Multiply $\frac{6}{1 - 7 x}$ by $\frac{1}{x}$ .

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

Step 1.2.2

Reorder factors in $\frac{6}{(1 - 7 x) x}$ .

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

Step 2

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

$x (1 - 7 x) = 0$

Step 3

Solve for $x$ .

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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 $0$ .

$x = 0$

$1 - 7 x = 0$

Step 3.2

Set $x$ equal to $0$ .

$x = 0$

Step 3.3

Set $1 - 7 x$ equal to $0$ and solve for $x$ .

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Step 3.3.1

Set $1 - 7 x$ equal to $0$ .

$1 - 7 x = 0$

Step 3.3.2

Solve $1 - 7 x = 0$ for $x$ .

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Step 3.3.2.1

Subtract $1$ from both sides of the equation.

$- 7 x = - 1$

Step 3.3.2.2

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

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Step 3.3.2.2.1

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

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

Step 3.3.2.2.2

Simplify the left side.

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Step 3.3.2.2.2.1

Cancel the common factor of $- 7$ .

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Step 3.3.2.2.2.1.1

Cancel the common factor.

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

Step 3.3.2.2.2.1.2

Divide $x$ by $1$ .

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

Step 3.3.2.2.3

Simplify the right side.

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Step 3.3.2.2.3.1

Dividing two negative values results in a positive value.

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

Step 3.4

The final solution is all the values that make $x (1 - 7 x) = 0$ true.

$x = 0, \frac{1}{7}$

Step 4

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

Interval Notation:

$(- \infty, 0) \cup (0, \frac{1}{7}) \cup (\frac{1}{7}, \infty)$

Set-Builder Notation:

${x | x \neq 0, \frac{1}{7}}$

Step 5