Finite Math Examples

$f (x) = \frac{1}{5} \cdot \tan (\frac{1}{4} x) + 2$

Step 1

Set the argument in $\tan (\frac{1}{4} x)$ equal to $\frac{π}{2} + π n$ to find where the expression is undefined.

$\frac{1}{4} x = \frac{π}{2} + π n$ , for any integer $n$

Step 2

Solve for $x$ .

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Step 2.1

Multiply both sides of the equation by $4$ .

$4 (\frac{1}{4} x) = 4 (\frac{π}{2} + π n)$

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 $4 (\frac{1}{4} x)$ .

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Step 2.2.1.1.1

Combine $\frac{1}{4}$ and $x$ .

$4 \frac{x}{4} = 4 (\frac{π}{2} + π n)$

Step 2.2.1.1.2

Cancel the common factor of $4$ .

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Step 2.2.1.1.2.1

Cancel the common factor.

$4 \frac{x}{4} = 4 (\frac{π}{2} + π n)$

Step 2.2.1.1.2.2

Rewrite the expression.

$x = 4 (\frac{π}{2} + π n)$

Step 2.2.2

Simplify the right side.

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Step 2.2.2.1

Simplify $4 (\frac{π}{2} + π n)$ .

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Step 2.2.2.1.1

Apply the distributive property.

$x = 4 \frac{π}{2} + 4 (π n)$

Step 2.2.2.1.2

Cancel the common factor of $2$ .

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Step 2.2.2.1.2.1

Factor $2$ out of $4$ .

$x = 2 (2) \frac{π}{2} + 4 (π n)$

Step 2.2.2.1.2.2

Cancel the common factor.

$x = 2 \cdot 2 \frac{π}{2} + 4 (π n)$

Step 2.2.2.1.2.3

Rewrite the expression.

$x = 2 π + 4 (π n)$

$x = 2 π + 4 π n$

Step 2.3

Reorder $2 π$ and $4 π n$ .

$x = 4 π n + 2 π$

Step 3

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

Set-Builder Notation:

${x | x \neq 4 π n + 2 π}$ , 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: ${x | x \neq 4 π n + 2 π}$ , for any integer $n$

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

Step 6