Algebra Examples

$y = 9 \cot (2 x - \frac{π}{2}) - 3$

Step 1

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

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

Step 2

Solve for $x$ .

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Step 2.1

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

$2 x = π n + \frac{π}{2}$

Step 2.2

Divide each term in $2 x = π n + \frac{π}{2}$ by $2$ and simplify.

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Step 2.2.1

Divide each term in $2 x = π n + \frac{π}{2}$ by $2$ .

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

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.

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

Step 2.2.2.1.2

Divide $x$ by $1$ .

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

Step 2.2.3

Simplify the right side.

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Step 2.2.3.1

Simplify each term.

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Step 2.2.3.1.1

Multiply the numerator by the reciprocal of the denominator.

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

Step 2.2.3.1.2

Multiply $\frac{π}{2} \cdot \frac{1}{2}$ .

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Step 2.2.3.1.2.1

Multiply $\frac{π}{2}$ by $\frac{1}{2}$ .

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

Step 2.2.3.1.2.2

Multiply $2$ by $2$ .

$x = \frac{π n}{2} + \frac{π}{4}$

Step 3

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

Set-Builder Notation:

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

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

Step 6