Chapter 8, Explorations 6-7

Use the sliders below to choose the left and right endpoints xmin and xmax of an input interval $$I$$ so that both endpoints have values less than $$\frac{1}{2}$$. The interval $$I$$ you have chosen is in blue and it's image under the map $$x_{n+1} = f_4(x_n)$$ is in red.

• How are the endpoints of the image interval $$f_4(I)$$ determined?
• For a point in the image interval, how many points in the domain interval $$I$$ get mapped onto that point?
Answer these questions again when xmin and xmax are both greater than $$\frac{1}{2}$$. Now take xmin $$< \frac{1}{2} <$$ xmax and again use the tool on the website to observe $$f_4(I)$$.
• How are endpoints of the image interval $$f_4(I)$$ determined?
• For a point in the image interval, how many points in the domain interval $$I$$ get mapped onto that point?
Note: You may choose xmin $$<$$ xmax and the applet will still work, but the labels may be confusing.

xmax:

xmin: