Intermediate Value Theorem
For all (red) y between f(a) and f(b) there exists c in [a,b] such that f(c)=y. Equivalently, the (blue) image im f=f([a,b]) contains the (red) interval J.
Slide c along the x-axis to find a value such that the blue dots f(c) cover any red dot (any y). Press the refresh button (top right) to start another experiment. Other options: 1. Slide a,b to change the interval under consideration. 2. Change f(x): refresh picture then enter in the input field another function definition, e.g., f(x)=x^2. (You may have to grab and move the image to shake off the blue dots from the previous experiment - click on the last button in the toolbar, move the image, then click on the first button in the toolbar).