"One-to-one" here implies that no particular y-value occurs more than once. Looking at the sloping part of the shown function, we see that this is the case. But not the horizontal parts of the graph, where y takes on one value multiple times (y=1 and y=4).
Apply the Horizontal Line Test. If a horiz. line drawn thru the graph intersects the graph in more than one place, the function fails this Test and is not one-to-one. So we conclude that the function shown is NOT one-to-one.
The range of this function is a list of all possible values that the function may take on. From the graph it is evident that the smallest y-value is almot but not quite -2, and that the largest is 4. Thus, the range is (-2,4).