Find the Domain and Range f(theta)=-5cot(2pitheta+pi)

$f (θ) = - 5 \cot (2 π θ + π)$

Step 1

Set the argument in $\cot (2 π θ + π)$ equal to $π n$ to find where the expression is undefined.

$2 π θ + π = π n$ , for any integer $n$

Step 2

Solve for $θ$ .

Subtract $π$ from both sides of the equation.

$2 π θ = π n - π$

Divide each term in $2 π θ = π n - π$ by $2 π$ and simplify.

Divide each term in $2 π θ = π n - π$ by $2 π$ .

$\frac{2 π θ}{2 π} = \frac{π n}{2 π} + \frac{- π}{2 π}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 π θ}{2 π} = \frac{π n}{2 π} + \frac{- π}{2 π}$

Rewrite the expression.

$\frac{π θ}{π} = \frac{π n}{2 π} + \frac{- π}{2 π}$

Cancel the common factor of $π$ .

Cancel the common factor.

$\frac{π θ}{π} = \frac{π n}{2 π} + \frac{- π}{2 π}$

Divide $θ$ by $1$ .

$θ = \frac{π n}{2 π} + \frac{- π}{2 π}$

Simplify the right side.

Simplify each term.

Cancel the common factor of $π$ .

Cancel the common factor.

$θ = \frac{π n}{2 π} + \frac{- π}{2 π}$

Rewrite the expression.

$θ = \frac{n}{2} + \frac{- π}{2 π}$

Cancel the common factor of $π$ .

Cancel the common factor.

$θ = \frac{n}{2} + \frac{- π}{2 π}$

Rewrite the expression.

$θ = \frac{n}{2} + \frac{- 1}{2}$

Move the negative in front of the fraction.

$θ = \frac{n}{2} - \frac{1}{2}$

Step 3

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

Set-Builder Notation:

${θ | θ \neq \frac{n}{2} - \frac{1}{2}}$ , 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: ${θ | θ \neq \frac{n}{2} - \frac{1}{2}}$ , for any integer $n$

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

Step 6

Find the Domain and Range f(theta)=-5cot(2pitheta+pi)

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