Find the Domain and Range f(x)=4cot(2x)+3

$f (x) = 4 \cot (2 x) + 3$

Step 1

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

$2 x = π n$ , for any integer $n$

Step 2

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

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

$\frac{2 x}{2} = \frac{π n}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x}{2} = \frac{π n}{2}$

Divide $x$ by $1$ .

$x = \frac{π n}{2}$

Step 3

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

Set-Builder Notation:

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

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

Step 6