Continuos Image Of A Closed Interval

If you adjust the end points, A, and B, then, as you drag "E" back and forth from A to B, the image of the interval after applying "f" is traced out in red on the y-axis. Notice that the image is a single, connected, closed interval. This means that it has a high point and a low point (the Max-Min theorem we'll expand on in Chapter 3), and every point in between (the Intermediate Value Theorem in this section.)

Set A at -2, and B at 2. Then f(A) is about 4 and f(B) is about -1. The IVT guarantees that if you pick any point on the y-axis between -1 and 2, there's at least one location for "E" with f(E) = that value.

Find two "E"s that correspond to a height of 3.

Find an "E" that corresponds to a height of 1.

Now, slide "A" and "B" to -1 and 4. What's the range of values guaranteed by the IVT?

Dave Matthews, Created with GeoGebra