Algebra Examples

$f (x) = 3 \tan (2 x - \frac{3 π}{2})$

Step 1

Set the argument in $\tan (2 x - \frac{3 π}{2})$ equal to $\frac{π}{2} + π n$ to find where the expression is undefined.

$2 x - \frac{3 π}{2} = \frac{π}{2} + π n$ , for any integer $n$

Step 2

Solve for $x$ .

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Step 2.1

Move all terms not containing $x$ to the right side of the equation.

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Step 2.1.1

Add $\frac{3 π}{2}$ to both sides of the equation.

$2 x = \frac{π}{2} + π n + \frac{3 π}{2}$

Step 2.1.2

Combine the numerators over the common denominator.

$2 x = π n + \frac{π + 3 π}{2}$

Step 2.1.3

Add $π$ and $3 π$ .

$2 x = π n + \frac{4 π}{2}$

Step 2.1.4

Cancel the common factor of $4$ and $2$ .

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Step 2.1.4.1

Factor $2$ out of $4 π$ .

$2 x = π n + \frac{2 (2 π)}{2}$

Step 2.1.4.2

Cancel the common factors.

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Step 2.1.4.2.1

Factor $2$ out of $2$ .

$2 x = π n + \frac{2 (2 π)}{2 (1)}$

Step 2.1.4.2.2

Cancel the common factor.

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

Step 2.1.4.2.3

Rewrite the expression.

$2 x = π n + \frac{2 π}{1}$

Step 2.1.4.2.4

Divide $2 π$ by $1$ .

$2 x = π n + 2 π$

Step 2.2

Divide each term in $2 x = π n + 2 π$ by $2$ and simplify.

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Step 2.2.1

Divide each term in $2 x = π n + 2 π$ by $2$ .

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

Step 2.2.2

Simplify the left side.

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Step 2.2.2.1

Cancel the common factor of $2$ .

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Step 2.2.2.1.1

Cancel the common factor.

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

Step 2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{π n}{2} + \frac{2 π}{2}$

Step 2.2.3

Simplify the right side.

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Step 2.2.3.1

Cancel the common factor of $2$ .

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Step 2.2.3.1.1

Cancel the common factor.

$x = \frac{π n}{2} + \frac{2 π}{2}$

Step 2.2.3.1.2

Divide $π$ by $1$ .

$x = \frac{π n}{2} + π$

Step 3

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

Set-Builder Notation:

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

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

Step 6