This post is an update of the post Demonstrating the inverse image of a function.

To manipulate the demos in this post, you must have Wolfram CDF Player installed on your computer. It is available free from the Wolfram website. CDF Player works on most desktop computers using Firefox, Safari and Internet Explorer, but not Chrome.

The code for the demos, with some explanatory remarks, is in the file InverseImage.nb on my ,Mathematica website. That website also includes some other examples as .cdf files.

If the diagrams don’t appear, or appear but show pink diagrams, or if the formulas in the text are too high or too low, refresh the screen.

• The vertical red interval has the horizontal green interval(s) as inverse image.
• You can move the sliders back and forth to move to different points on the curve. The sliders control the vertical red interval. $a$ is the lower point of the vertical red line and $b$ is the upper point.
• As you move the sliders back and forth you will see the inverse image breaking up into a disjoint union in intervals, merging into a single interval, or disappearing entirely.
• The arrow at the upper right makes it run automatically.
• If you are using Mathematica, you can enter values into the boxes, but if you are using CDF Player, you can only change the number using the slider or the plus and minus incrementers.

 

This is the graph of $y=x^2-1$.

The graph of $-.5 + .5 x + .2 x^2 – .19 x^3 – .015 x^4 + .01 x^5 $

The graph of the rational function $0.5 x+\frac{1.5 \left(x^4-1\right)}{x^4+1}$

The graph of a straight line whose slope can be changed. You can design demos of other functions with variable parameters.

The graph of the sine function. The other demos were coded using the Mathematica Reduce function to get the inverse image. This one had to be done in an ad hoc way as explained in the InverseImage.nb file.