Find the Domain and Range y=5cot(x/3-pi/4)

$y = 5 \cot (\frac{x}{3} - \frac{π}{4})$

Step 1

Set the argument in $\cot (\frac{x}{3} - \frac{π}{4})$ equal to $π n$ to find where the expression is undefined.

$\frac{x}{3} - \frac{π}{4} = π n$ , for any integer $n$

Step 2

Solve for $x$ .

Add $\frac{π}{4}$ to both sides of the equation.

$\frac{x}{3} = π n + \frac{π}{4}$

Multiply both sides of the equation by $3$ .

$3 \frac{x}{3} = 3 (π n + \frac{π}{4})$

Simplify both sides of the equation.

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$3 \frac{x}{3} = 3 (π n + \frac{π}{4})$

Rewrite the expression.

$x = 3 (π n + \frac{π}{4})$

Simplify the right side.

Simplify $3 (π n + \frac{π}{4})$ .

Apply the distributive property.

$x = 3 (π n) + 3 \frac{π}{4}$

Combine $3$ and $\frac{π}{4}$ .

$x = 3 π n + \frac{3 π}{4}$

Step 3

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

Set-Builder Notation:

${x | x \neq 3 π n + \frac{3 π}{4}}$ , 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: ${x | x \neq 3 π n + \frac{3 π}{4}}$ , for any integer $n$

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

Step 6