Question

How do you find the domain & range for `(1/2)tan(x)`?

Answer 1

Domain: `\mathbb{R} \setminus {pi/2+kpi,\ k \in \mathbb{Z} }`

Range: `(-infty, infty)`.

Explanation:

The range is the same of the tangent, because if it is legit to compute `tan(x)`, then dividing that value by `2` is surely ok.

So, the domain is the real set minus the points in which `cos(x)=0`, i.e.

`\mathbb{R} \setminus {pi/2+kpi,\ k \in \mathbb{Z} }`

The same goes for the range: since the tangent ranges from `-infty` to `infty`, half of this range is still `(-infty, infty)`.

Note that this is true only because the interval is infinite in both direction, if the original range of `f(x)` is `[a,b]`, then the range of `f(x)/2` is `[a/2, b/2]`.