{"id":378,"date":"2014-12-13T20:23:37","date_gmt":"2014-12-14T00:23:37","guid":{"rendered":"http:\/\/advancedmathyoungstudents.com\/blog\/?p=378"},"modified":"2014-12-13T20:23:37","modified_gmt":"2014-12-14T00:23:37","slug":"those-flipping-inverse-graphs","status":"publish","type":"post","link":"https:\/\/advancedmathyoungstudents.com\/blog\/?p=378","title":{"rendered":"Those Flipping Inverse Graphs"},"content":{"rendered":"
\"Sometimes the sophisticated thing is easier to understand \r\nthan the kludgey thing. The flipped plots in the books were \r\ndriving me crazy. They don't even pass the dimensional analysis \r\ntest.\"\r\n -- John Denker<\/pre>\n
I have been known to rant occasionally about some of the curriculum choices we have made for high school mathematics.\u00a0 I feel the math we teach should be useful or interesting, and ideally both.\u00a0 But it should also be true!\u00a0 I think of that as the teacher’s equivalent of medical rule: first, do no harm.<\/p>\n
What brings this to mind is an email discussion I have been having with Dr. John Denker, a physicist who I have “met” through the Phys-L.org email list.\u00a0 He has, as you see above, some very reasonable objections to the way that my book repeatedly points out the symmetry between the graph of a function and its inverse.\u00a0 I would like to discuss his objections here in this post\u00a0and also to show his suggested alternative. Then, in the spirit of this blog,\u00a0our email discussion led me to\u00a0remember a nice\u00a0method for\u00a0using graphs to evaluate iterated functions.\u00a0 I will write about that in another post soon.<\/p>\n
[By the way, to any current or future physics teachers out there:\u00a0I\u00a0strongly recommend joining\u00a0the\u00a0email list Phys-L.\u00a0 It may be the single most helpful resource\u00a0I know of, populated by\u00a0many patient and enthusiastic experts, including Dr. Denker.\u00a0 My current students can attest to the many\u00a0occasions when my teaching has\u00a0been assisted and influenced by\u00a0ideas I\u00a0encountered on Phys-L.]<\/p>\n
OK, on to those flipping graphs…<\/p>\n
CLAIM:\u00a0 If you graph a function and its inverse on the same axes, the two graphs will have an interesting and esthetically pleasing symmetry with respect to the line y = x.<\/strong><\/p>\n
I am by no means the only one who makes this claim!\u00a0 Search the web for “inverse function graph symmetry” and you will find that claim stated quite frequently.<\/p>\n
For example, consider the squaring function, with its domain restricted to the non-negative real numbers.\u00a0 It has an inverse function, the square root function, and sure enough, if you graph both functions on the same axes, you see the symmetry that I am talking about:<\/p>\n
\"post24pic3\"<\/a><\/p>\n
The graph above was made at www.desmos.com<\/a>, a very user-friendly graphing site.\u00a0 And the graph certainly provides evidence in favor of the claim.\u00a0 And I personally find this to be, as I said, esthetically pleasing.\u00a0\u00a0Also, this idea\u00a0is not without application.\u00a0 When you learn\u00a0a formula for\u00a0finding derivatives of inverse functions, this symmetry will present a graphical way to understand\u00a0that formula.\u00a0 Still, if I had to guess, I suspect that most of the people who write about this claim do it\u00a0because they\u00a0like how the graphs look.\u00a0 They look nice.<\/p>\n
OBJECTION #1<\/strong><\/p>\n
“If you graph a function and its inverse…”\u00a0\u00a0But why would you ever\u00a0do that?\u00a0 As JD pointed out to me:<\/p>\n
I don't think anybody ever needs to flip the axis to form \r\nthe inverse function. Just read the axes in the other order.<\/pre>\n
That’s a good point.\u00a0 For example, let’s look again at the squaring function by itself this time.<\/p>\n
\"post24pic5\"<\/a><\/p>\n
The graph is a convenient way to represent the relationship between the length of the side of a square and its corresponding area.\u00a0 But if you already knew the area and wanted the length, you would not need to draw a new graph!\u00a0 You would just need to know how to read the graph you already have!\u00a0 You’d start on the vertical, area axis, go over to the curve and then read down to the horizontal axis where you would read the length.\u00a0 Same graph, different purpose.<\/p>\n
 <\/p>\n
BUT STILL…<\/p>\n
OK, I don’t need<\/em> to graph the inverse function.\u00a0 But what if I just want to because I like how it looks.\u00a0 Is it wrong?\u00a0 Does it violate the teacher’s version of the Hippocratic Oath?\u00a0\u00a0Alas, not always, but often, and in a way that I should have\u00a0considered, seeing as a major theme in my book is\u00a0how math is actually used in science.\u00a0 My current students may find it amusing that\u00a0what I have overlooked are the units!\u00a0\u00a0I may\u00a0feel like graphing the two functions ON THE SAME AXES, but in most cases, it would\u00a0be wrong.\u00a0 Which brings us to…<\/p>\n
OBJECTION #2<\/strong><\/p>\n
The flipped graphs do not have the right units on each axis!<\/em><\/p>\n
You don’t need to use sophisticated physics or chemistry to show this.\u00a0 Our first\u00a0graph above with\u00a0the squaring function and the square root function illustrates the problem sufficiently.\u00a0 Look again at the vertical axis.\u00a0 Pick a number.\u00a0 OK, 2.\u00a0\u00a0But\u00a02 WHAT?\u00a0Is it inches\u00a0or square inches? Meters or square meters?\u00a0 D’oh.<\/p>\n
The bottom line is that if your functions represent physical quantities, then those quantities have units.\u00a0 And unless the units happen to be the same, then\u00a0you cannot just flip the graph.\u00a0 You have to flip the axes as well.\u00a0You just can’t throw those graphs on the same axes.\u00a0 Look at them side-by-side:<\/p>\n
\"post24pic6\"<\/a><\/p>\n
If you want to show the symmetry of the original claim, you could take these two graphs, put them on overhead transparencies and lay one on top of the other:<\/p>\n
\"post24pic7\"<\/a><\/p>\n
It’s not as pretty, but at least its not wrong.<\/p>\n
When you are focusing on the math, it is easy to forget that these\u00a0quantities have physical meaning and units that go with them.\u00a0 For example, there are many books and websites where\u00a0you\u00a0can see a function and its derivative graphed on the same axes.\u00a0 (It’s even done that way on some of the websites that I used in earlier posts about derivatives.)\u00a0 But that makes no sense!\u00a0 For example, how can you graph position and velocity on the same axes?\u00a0 They don’t have the same units.<\/p>\n
I did a quick web search for “function derivative graph applet”.\u00a0 Here is\u00a0one site (of many)\u00a0I found that puts the function and derivative on the same axes:<\/p>\n
http:\/\/www.math.uri.edu\/~bkaskosz\/flashmo\/derplot\/<\/a><\/p>\n
\"post24pic2\"<\/a><\/p>\n
Here’s one that\u00a0correctly puts them on different axes:<\/p>\n
http:\/\/www.flashandmath.com\/advanced\/derdraw\/DerivativeDraw.html<\/a><\/p>\n
\"post24pic1\"<\/a><\/p>\n
These are both very\u00a0clever websites, good for developing a feel for what a function’s derivative looks like.\u00a0 But the second one is better — it remembers that these are quantities with units.\u00a0 (Also, the shading on the derivative graph is a neat bit of fore-shadowing.\u00a0 That’s a topic for another upcoming post as well.)<\/p>\n
Getting back to the Flipping Graphs…<\/strong><\/p>\n
If you don’t like the looks of the two overlaid transparencies, there are some\u00a0other ways to visualize these functions and their inverses.\u00a0 John Denker presents a few of them on his website here:<\/p>\n
https:\/\/www.av8n.com\/physics\/math-intro.htm#ex-graphs<\/a><\/p>\n
Here is one diagram (figure 10)\u00a0from his site that I especially liked:<\/p>\n
\"post24pic4\"<\/a><\/p>\n
The “reflector panel” reminded me of another clever method of interpreting graphs.\u00a0 More on that in a future post…<\/p>\n
 <\/p>\n","protected":false},"excerpt":{"rendered":"
“Sometimes the sophisticated thing is easier to understand than the kludgey thing. The flipped plots in the books were driving me crazy. They don’t even pass the dimensional analysis test.” — John Denker I have been known to rant occasionally … Continue reading →<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"jetpack_post_was_ever_published":false},"categories":[1],"tags":[],"jetpack_featured_media_url":"","jetpack_shortlink":"https:\/\/wp.me\/p4uvY7-66","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/advancedmathyoungstudents.com\/blog\/index.php?rest_route=\/wp\/v2\/posts\/378"}],"collection":[{"href":"https:\/\/advancedmathyoungstudents.com\/blog\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/advancedmathyoungstudents.com\/blog\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/advancedmathyoungstudents.com\/blog\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/advancedmathyoungstudents.com\/blog\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=378"}],"version-history":[{"count":10,"href":"https:\/\/advancedmathyoungstudents.com\/blog\/index.php?rest_route=\/wp\/v2\/posts\/378\/revisions"}],"predecessor-version":[{"id":395,"href":"https:\/\/advancedmathyoungstudents.com\/blog\/index.php?rest_route=\/wp\/v2\/posts\/378\/revisions\/395"}],"wp:attachment":[{"href":"https:\/\/advancedmathyoungstudents.com\/blog\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=378"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/advancedmathyoungstudents.com\/blog\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=378"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/advancedmathyoungstudents.com\/blog\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=378"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}