Question

please help me figure out how to determine the range of the following graph (the line y=5 is a horizontal asymptote)

Answer 1

we are given the graph of the function and we are interested in finding the range of the function. Recall that the range of a graph is simply the set of values on the y axis, for which there is a point on the graph that has that y coordinate.

One easy way to spot this set, is by taking any point on the graph and then drawing a horizontal line. Wherever the line crosses the y axis, that point is included in the range.

From the graph, we can see that no part of the graph has values with y coordinate less than 5. That is, any number less than 5 in the y coordinate would indicate that there is no point on the graph at that "height". So every number less than 5 is excluded from the range.

We are also told that line y=5 is a horizontal asymptote. This means that despite the graph is really close to the line y=5 (and it keeps getting closer and closer as x increases), it never touches the line. This means that the point 5 is excluded from the range.

Finally, we can see that above the horizontal line y=5, if we draw a horizontal line on the graph, it will touch the y axis. This means that every number greater than 5 is part of the range. Then, the set of numbers that represent the range is

$(5,\text{infinity)}$