Function y=tan(x) is not invertible (not one-to-one), because it fails horizontal line test. We can draw such horizontal line that it will intersect graph of `y=tan(x)` more than once.
However, if we consider interval `(-pi/2,pi/2)` then on this interval function passes horizontal line test. On this interval function is increasing and takes all values. Therefore, for function `y=tan(x),-pi/2<x<pi/2` there exists inverse function. This function is denoted by `y=arctan(x)` (sometimes also denoted by `y=atan(x)` or `y=tan^(-1)(x)`) and is read as arctangent.
We can obtain graph of the function `y=arctan(x)` from graph of the function `y=tan(x),-pi/2<x<pi/2` by symmetrically transforming it about line y=x (see figure).
Properties of the function `y=arctan(x)`:
- Domain is all number line.
- Range is interval `(-pi/2,pi/2)`.
- Function is odd: `arctan(-x)=-arctan(x)`.
- Function is increasing.
- Lines `y=pi/2` and `y=-pi/2` are horizontal asymptotes as `x->+oo` and`x->-oo` respectively.
From above it follows that records `y=arctan(x)` and `x=tan(y),-pi/2<y<pi/2` are equivalent. If we substitute expression for y from first equation into second then we will obtain that `x=tan(arctan(x))`.
Therefore, for any x`tan(arctan(x))=x,-pi/2<arctan(x)<pi/2`.