Finite Math Examples

$f (x) = \frac{3}{| 8 x - 5 |}$

Step 1

Set the denominator in $\frac{3}{| 8 x - 5 |}$ equal to $0$ to find where the expression is undefined.

$| 8 x - 5 | = 0$

Step 2

Solve for $x$ .

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Step 2.1

Remove the absolute value term. This creates a $\pm$ on the right side of the equation because $| x | = \pm x$ .

$8 x - 5 = \pm 0$

Step 2.2

Plus or minus $0$ is $0$ .

$8 x - 5 = 0$

Step 2.3

Add $5$ to both sides of the equation.

$8 x = 5$

Step 2.4

Divide each term in $8 x = 5$ by $8$ and simplify.

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Step 2.4.1

Divide each term in $8 x = 5$ by $8$ .

$\frac{8 x}{8} = \frac{5}{8}$

Step 2.4.2

Simplify the left side.

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Step 2.4.2.1

Cancel the common factor of $8$ .

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Step 2.4.2.1.1

Cancel the common factor.

$\frac{8 x}{8} = \frac{5}{8}$

Step 2.4.2.1.2

Divide $x$ by $1$ .

$x = \frac{5}{8}$

Step 3

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

Interval Notation:

$(- \infty, \frac{5}{8}) \cup (\frac{5}{8}, \infty)$

Set-Builder Notation:

${x | x \neq \frac{5}{8}}$

Step 4

The range is the set of all valid $y$ values. Use the graph to find the range.

Interval Notation:

$(0, \infty)$

Set-Builder Notation:

${y | y > 0}$

Step 5

Determine the domain and range.

Domain: $(- \infty, \frac{5}{8}) \cup (\frac{5}{8}, \infty), {x | x \neq \frac{5}{8}}$

Range: $(0, \infty), {y | y > 0}$

Step 6