We know that is defined in terms of the exponential function as follows
 
 

By rewriting the function as
 
 

and noting that both factors in the numerator, as well as the denominator are positive for all values of x, we have that the graph of is always positive (in fact, it is always > = 1).

Now, the first derivative of is given by
 
 

which may rewritten as
 
 

So, since is always positive, and the denominator (=  2)  of is always positive, we have that
 
 

So, the graph of the function is decreasing when x < 0 and increasing when x > 0.
 

Now, we take the second derivative of and obtain
 
 

we have already noted that  is positive for all values of x, so the graph is always concave up.