Find the Domain and Range tan(arctan(x))

Set the argument in ${\displaystyle \tan \left(\arctan \left(x\right)\right)}$ equal to ${\displaystyle \frac{\pi }{2}+\pi n}$ to find where the expression is undefined.

${\displaystyle \arctan \left(x\right)=\frac{\pi }{2}+\pi n}$, for any integer ${\displaystyle n}$

Take the inverse arctangent of both sides of the equation to extract ${\displaystyle x}$ from inside the arctangent.

The domain is all values of ${\displaystyle x}$ that make the expression defined.

Set-Builder Notation:

${\displaystyle \left\{x\right|x\ne \tan (\frac{\pi }{2}+\pi n)\}}$, for any integer ${\displaystyle n}$

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

Interval Notation:

Set-Builder Notation:

Determine the domain and range.

Domain: ${\displaystyle \left\{x\right|x\ne \tan (\frac{\pi }{2}+\pi n)\}}$, for any integer ${\displaystyle n}$

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