Finite Math Examples

$f (x) = 4 x$ , $g (x) = 3 x + 1$

Step 1

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

$\frac{4 x}{3 x + 1}$

Step 2

Set the denominator in $\frac{4 x}{3 x + 1}$ equal to $0$ to find where the expression is undefined.

$3 x + 1 = 0$

Step 3

Solve for $x$ .

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Step 3.1

Subtract $1$ from both sides of the equation.

$3 x = - 1$

Step 3.2

Divide each term in $3 x = - 1$ by $3$ and simplify.

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Step 3.2.1

Divide each term in $3 x = - 1$ by $3$ .

$\frac{3 x}{3} = \frac{- 1}{3}$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $3$ .

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Step 3.2.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{- 1}{3}$

Step 3.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 1}{3}$

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

Move the negative in front of the fraction.

$x = - \frac{1}{3}$

Step 4

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

Interval Notation:

$(- \infty, - \frac{1}{3}) \cup (- \frac{1}{3}, \infty)$

Set-Builder Notation:

${x | x \neq - \frac{1}{3}}$

Step 5