Find the Domain and Range f(x)=cot(1/3x)

$f (x) = \cot (\frac{1}{3} x)$

Set the argument in $\cot (\frac{1}{3} x)$ equal to $π n$ to find where the expression is undefined.

$\frac{1}{3} x = π n$ , for any integer $n$

Solve for $x$ .

Multiply both sides of the equation by $3$ .

$3 (\frac{1}{3} x) = 3 (π n)$

Simplify both sides of the equation.

Simplify the left side.

Simplify $3 (\frac{1}{3} x)$ .

Combine $\frac{1}{3}$ and $x$ .

$3 \frac{x}{3} = 3 (π n)$

Cancel the common factor of $3$ .

Cancel the common factor.

$3 \frac{x}{3} = 3 (π n)$

Rewrite the expression.

$x = 3 (π n)$

Simplify the right side.

Remove parentheses.

$x = 3 π n$

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

Set-Builder Notation:

${x | x \neq 3 π n}$ , for any integer $n$

The range is the set of all valid $y$ values. Use the graph to find the range.

Interval Notation:

$(- \infty, \infty)$

Set-Builder Notation:

${y | y \in ℝ}$

Determine the domain and range.

Domain: ${x | x \neq 3 π n}$ , for any integer $n$

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