Unit G Summary Question #10: Rang of a Rational Function.....

10. While the domain of a rational function depends on DIVAH, what do you think the range of a rational function depends on. Give an example.

NOTE: D-omain
               I-s the
              V-ertical
              A-asymptote & the
              H-ole

I believe that range of a rational function is dependent upon all of the y-values that belong to the x-values that come from the vertical asymptotes and holes. For example, the vertical asyptotes for a graph were to be x=-2 and x=2 and the hole would be x=1. This would make the  the domain, or the bad values, -2,2, and 1. With that, the range would be the infinite numbers that these numbers passed from +infinity (the + numbers of the y-axis) and - infinity (the - numbers of the y-axis). Because the lines run vertically downwards, the coordinate pairs would be something like (-2,4), (-2,3), (-2,2) ,(-2,1) ,(-2,0) ,(-2,-1) ,(-2,-2), (-2,-3) and so on upwards and down the y-axis, with the domain staying the same. This would be the same for the x=2 vertical asymptote. In short, the range would be ALL REAL NUMBERS. With that, the range of a hole would simply be the y-value. For exapmle, the hole would have the cordinates of (1, -1/2), -1/2 would be the range.

As you can see in this image, the vertical asymptote is
x=-1. The range, or y-values, run up and down make it
ALL REAL NUMBERS.

On the other hand, if the y-value was a horizontal asymptote, the line would run straight across the y-axis and so it would NOT be all real numbers. For example, if y=2, the y-value would remain constant whereas you x-value would be constantly changing as you traveled across the graph from left to right. [(1,2),(2,2),(3,2),(4,2),(5,2), etc.,...] thus, the range of a horizontal asymptote would simply be the value of y. We have already been taking notice of this and it is noted in our limit notation. (If needed: Review the lesson in Unit G Summary Question #2).