Given the function graphed below, find the domain and range of Note there is an invisible vertical line at and the two arms of the graph are extending downward (and upward) forever, getting arbitrarily close to that vertical line, but never touching it. Also note that the two arms extend forever to the left and right, getting arbitrarily close to the -axis, but never touching it.
The domain of is
and the range of is
.
To find the domain, we try to visualize all of the -values that are valid inputs for this function. The arrows pointing left and right on the curve indicate that whatever pattern we see in the graph continues off to the left and right. So for -values far to the right or left, we will be able to get an output for
The arrows pointing up and down are supposed to indicate that the curve will get closer and closer to the vertical line after the curve leaves the viewing window we are using. So even when is some number very close to we will be able to get an output for
The one -value that doesn’t behave is If we tried to use that as an input, there is no point on the graph directly above or below that on the -axis. So the domain is
To find the range, we try to visualize all of the -values that are possible outputs for this function. Sliding the ink of the curve left/right onto the -axis reveals that is the only -value that we could never obtain as an output. So the range is