How would you describe long term behavior?

# Review problem 3-C

I'm not quite sure what you're asking but I'm guessing you'd like to know how you, as a student taking the quiz, would answer problem 3-C.

Remember that in our introduction to the basics of iteration, we saw that there are several potential, long-term behaviors of an orbit generated by iteration. Among other possibilities, we saw that an orbit might be constant, it might converge to a constant, it might cycle between several values, etc. The question asks you to decide which of these possibilities happens in this case.

One geometric tool that can help decide which possibility actually occurs is cobweb plot. For this particular function, the cobweb plot ends up looking something like so:

It looks to me like (in the long-term) this orbit converges to the fixed point around $0.7$. Note that we might expect this to happen because it sure *looks* like the slope of the function at the point of intersection is less than one in absolute value.