Finite Math Examples

$f (x) = - 9 \csc (\frac{π}{3} x)$

Step 1

Set the argument in $\csc (\frac{π}{3} x)$ equal to $π n$ to find where the expression is undefined.

$\frac{π}{3} x = π n$ , for any integer $n$

Step 2

Solve for $x$ .

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Step 2.1

Multiply both sides of the equation by $\frac{3}{π}$ .

$\frac{3}{π} (\frac{π}{3} x) = \frac{3}{π} (π n)$

Step 2.2

Simplify both sides of the equation.

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Step 2.2.1

Simplify the left side.

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Step 2.2.1.1

Simplify $\frac{3}{π} (\frac{π}{3} x)$ .

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Step 2.2.1.1.1

Combine $\frac{π}{3}$ and $x$ .

$\frac{3}{π} \cdot \frac{π x}{3} = \frac{3}{π} (π n)$

Step 2.2.1.1.2

Cancel the common factor of $3$ .

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Step 2.2.1.1.2.1

Cancel the common factor.

$\frac{3}{π} \cdot \frac{π x}{3} = \frac{3}{π} (π n)$

Step 2.2.1.1.2.2

Rewrite the expression.

$\frac{1}{π} (π x) = \frac{3}{π} (π n)$

Step 2.2.1.1.3

Cancel the common factor of $π$ .

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Step 2.2.1.1.3.1

Factor $π$ out of $π x$ .

$\frac{1}{π} (π (x)) = \frac{3}{π} (π n)$

Step 2.2.1.1.3.2

Cancel the common factor.

$\frac{1}{π} (π x) = \frac{3}{π} (π n)$

Step 2.2.1.1.3.3

Rewrite the expression.

$x = \frac{3}{π} (π n)$

Step 2.2.2

Simplify the right side.

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Step 2.2.2.1

Cancel the common factor of $π$ .

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Step 2.2.2.1.1

Factor $π$ out of $π n$ .

$x = \frac{3}{π} (π (n))$

Step 2.2.2.1.2

Cancel the common factor.

$x = \frac{3}{π} (π n)$

Step 2.2.2.1.3

Rewrite the expression.

$x = 3 n$

Step 3

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

Set-Builder Notation:

${x | x \neq 3 n}$ , for any integer $n$

Step 4

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

Interval Notation:

$(- \infty, - 9] \cup [9, \infty)$

Set-Builder Notation:

${y | y \leq - 9, y \geq 9}$

Step 5

Determine the domain and range.

Domain: ${x | x \neq 3 n}$ , for any integer $n$

Range: $(- \infty, - 9] \cup [9, \infty), {y | y \leq - 9, y \geq 9}$

Step 6