Finite Math Examples

$f (x) = - 6 x - 1$ , $g (x) = 4 x^{2}$

Step 1

Find the quotient of the functions.

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Step 1.1

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

$\frac{- 6 x - 1}{4 x^{2}}$

Step 1.2

Simplify.

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Step 1.2.1

Factor $- 1$ out of $- 6 x$ .

$\frac{- (6 x) - 1}{4 x^{2}}$

Step 1.2.2

Rewrite $- 1$ as $- 1 (1)$ .

$\frac{- (6 x) - 1 (1)}{4 x^{2}}$

Step 1.2.3

Factor $- 1$ out of $- (6 x) - 1 (1)$ .

$\frac{- (6 x + 1)}{4 x^{2}}$

Step 1.2.4

Simplify the expression.

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Step 1.2.4.1

Rewrite $- (6 x + 1)$ as $- 1 (6 x + 1)$ .

$\frac{- 1 (6 x + 1)}{4 x^{2}}$

Step 1.2.4.2

Move the negative in front of the fraction.

$- \frac{6 x + 1}{4 x^{2}}$

Step 2

Set the denominator in $\frac{6 x + 1}{4 x^{2}}$ equal to $0$ to find where the expression is undefined.

$4 x^{2} = 0$

Step 3

Solve for $x$ .

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Step 3.1

Divide each term in $4 x^{2} = 0$ by $4$ and simplify.

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Step 3.1.1

Divide each term in $4 x^{2} = 0$ by $4$ .

$\frac{4 x^{2}}{4} = \frac{0}{4}$

Step 3.1.2

Simplify the left side.

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Step 3.1.2.1

Cancel the common factor of $4$ .

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Step 3.1.2.1.1

Cancel the common factor.

$\frac{4 x^{2}}{4} = \frac{0}{4}$

Step 3.1.2.1.2

Divide $x^{2}$ by $1$ .

$x^{2} = \frac{0}{4}$

Step 3.1.3

Simplify the right side.

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Step 3.1.3.1

Divide $0$ by $4$ .

$x^{2} = 0$

Step 3.2

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{0}$

Step 3.3

Simplify $\pm \sqrt{0}$ .

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Step 3.3.1

Rewrite $0$ as $0^{2}$ .

$x = \pm \sqrt{0^{2}}$

Step 3.3.2

Pull terms out from under the radical, assuming positive real numbers.

$x = \pm 0$

Step 3.3.3

Plus or minus $0$ is $0$ .

$x = 0$

Step 4

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

Interval Notation:

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

Set-Builder Notation:

${x | x \neq 0}$

Step 5