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

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

Step 1

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

$4 x = π n$ , for any integer $n$

Step 2

Divide each term in $4 x = π n$ by $4$ and simplify.

Divide each term in $4 x = π n$ by $4$ .

$\frac{4 x}{4} = \frac{π n}{4}$

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 x}{4} = \frac{π n}{4}$

Divide $x$ by $1$ .

$x = \frac{π n}{4}$

Step 3

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

Set-Builder Notation:

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

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

Step 6