# Function y=arctan(x)

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):

1. Domain is all number line.
2. Range is interval (-pi/2,pi/2).
3. Function is odd: arctan(-x)=-arctan(x).
4. Function is increasing.
5. Lines y=pi/2 and y=-pi/2 are horizontal asymptotes as x->+oo andx->-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 xtan(arctan(x))=x,-pi/2<arctan(x)<pi/2.