Observe the graph of :

The function is a piecewise function.

I/D test :

If on the interval, then is increasing on the interval.

If on the interval, then is decreasing on the interval.

Observe the graph.

over the intervals , and .

Therefore, the function is increasing over the intervals , and .

The function in the interval .

Since , in the interval , where is a constant.

The function in the interval .

Since , in the interval , where is a constant.

The function in the interval .

Since , in the interval , where is a constant.

Here constant for the continuity of .  (Since is continuous in )

Graph :

Graph the function such that above all the conditions are satisfied :

.

Since , in the interval .

Since , in the interval .

Since , in the interval .

Therefore, the function exist everywhere except .

Graph :

Graph the function such that above all the conditions are satisfied :

.

Graph of the function :

.

Graph of the function : 

.