\((-\infty , \infty )\) is very long.

There is the left tail, which is the part with very very very very large negative numbers. There is the right tail, which is the part with very very very very large positive numbers. Then, there is the middle part.

All of the individual characteristics of a function happen in the middle.

That is, each function has its own middle bubble (subset of the domain) where all of the important locations live. The middle bubble contains the zeros, the discontinuities, the singularities, the critical numbers, the maximums, the minimums, the everything. Once we are outside the middle part of the domain, the function settles down into a consistent pattern of behavior and maintains that predictable behavior all the way to \(-\infty \) or \(\infty \) (in the domain).

This settled down pattern in the tails is called the end-behavior of the function.

Graphs use graphical symbols to help describe this end-behavior.

Here is the complete graph of the function \(G(x)\).

In the graph above, we see a vertical asymptote (singularity). This is inside the middle bubble. It is not a part of end-behavior.

In the graph above, we see an intercept (zero). This is inside the middle bubble. It is not a part of end-behavior.

The horizontal asymptote is communicating end-behavior. It tells us that as we move further out in the domain, on either side, the function approaches the value \(2\).

Here is the complete graph of the function \(H(t)\).

The graph of \(H\) does not have horizontal asymptote. That is meaningful. The fact that the horizontal asymptote is not included in the drawing tells us that the function does not settle down, getting closer to \(2\). Otherwise, there would have been a horizontal asymptote. (This is an example of a missing symbol communicating meaning.)

From this, we deduce that as we move out in the domain towards \(\infty \) (the right tail), the value of \(H\) grows bigger and bigger negatively.

We also deduce that as we move out in the domain towards \(-\infty \) (the left tail), the value of \(H\) grows bigger and bigger positively.

The graphs suggests that this unbounded growth might be very slow, but \(H\) does become unbounded.

Here is a complete graph of \(K(v)\).

The domain of the function \(K\) is \([-6, \infty )\). Therefore, it only has one end-behavior. As we move out in the domain towards \(\infty \), \(K\) approaches \(2\). The value of \(K\) continues to oscillate above and below \(2\) as it moves cloaser and closer to the value of \(2\).

\(K\) does not have an end-behavior as the domain moves towards \(-\infty \), because the domain doesn’t move towards \(-\infty \) (no left tai).

Here is a complete graph of \(p(x)\).

The domain of the function \(p\) is \([-6, 10]\). Therefore, it has no end-behavior. The domain doesn’t move towards \(-\infty \) or \(\infty \). The domain is bounded.