Precalculus Examples

$y = \tan (\frac{π}{5} x)$

Step 1

Set the argument in $\tan (\frac{π}{5} x)$ equal to $\frac{π}{2} + π n$ to find where the expression is undefined.

$\frac{π}{5} x = \frac{π}{2} + π n$ , for any integer $n$

Step 2

Solve for $x$ .

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Step 2.1

Multiply both sides of the equation by $\frac{5}{π}$ .

$\frac{5}{π} (\frac{π}{5} x) = \frac{5}{π} (\frac{π}{2} + π n)$

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 $\frac{5}{π} (\frac{π}{5} x)$ .

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Step 2.2.1.1.1

Combine $\frac{π}{5}$ and $x$ .

$\frac{5}{π} \cdot \frac{π x}{5} = \frac{5}{π} (\frac{π}{2} + π n)$

Step 2.2.1.1.2

Cancel the common factor of $5$ .

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Step 2.2.1.1.2.1

Cancel the common factor.

$\frac{5}{π} \cdot \frac{π x}{5} = \frac{5}{π} (\frac{π}{2} + π n)$

Step 2.2.1.1.2.2

Rewrite the expression.

$\frac{1}{π} (π x) = \frac{5}{π} (\frac{π}{2} + π n)$

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 $π x$ .

$\frac{1}{π} (π (x)) = \frac{5}{π} (\frac{π}{2} + π n)$

Step 2.2.1.1.3.2

Cancel the common factor.

$\frac{1}{π} (π x) = \frac{5}{π} (\frac{π}{2} + π n)$

Step 2.2.1.1.3.3

Rewrite the expression.

$x = \frac{5}{π} (\frac{π}{2} + π n)$

Step 2.2.2

Simplify the right side.

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Step 2.2.2.1

Simplify $\frac{5}{π} (\frac{π}{2} + π n)$ .

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Step 2.2.2.1.1

Apply the distributive property.

$x = \frac{5}{π} \cdot \frac{π}{2} + \frac{5}{π} (π n)$

Step 2.2.2.1.2

Cancel the common factor of $π$ .

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Step 2.2.2.1.2.1

Cancel the common factor.

$x = \frac{5}{π} \cdot \frac{π}{2} + \frac{5}{π} (π n)$

Step 2.2.2.1.2.2

Rewrite the expression.

$x = 5 (\frac{1}{2}) + \frac{5}{π} (π n)$

Step 2.2.2.1.3

Combine $5$ and $\frac{1}{2}$ .

$x = \frac{5}{2} + \frac{5}{π} (π n)$

Step 2.2.2.1.4

Cancel the common factor of $π$ .

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Step 2.2.2.1.4.1

Factor $π$ out of $π n$ .

$x = \frac{5}{2} + \frac{5}{π} (π (n))$

Step 2.2.2.1.4.2

Cancel the common factor.

$x = \frac{5}{2} + \frac{5}{π} (π n)$

Step 2.2.2.1.4.3

Rewrite the expression.

$x = \frac{5}{2} + 5 n$

Step 2.3

Reorder $\frac{5}{2}$ and $5 n$ .

$x = 5 n + \frac{5}{2}$

Step 3

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

Set-Builder Notation:

${x | x \neq 5 n + \frac{5}{2}}$ , 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:

$(- \infty, \infty)$

Set-Builder Notation:

${y | y \in ℝ}$

Step 5

Determine the domain and range.

Domain: ${x | x \neq 5 n + \frac{5}{2}}$ , for any integer $n$

Range: $(- \infty, \infty), {y | y \in ℝ}$

Step 6