inverse function command

[DrD]

Function Example:
y:=1/(sqrt(2*x-3));

Needed is a built in command to essentially do this:
simplify(solve(x = (1/(sqrt(2*y-3))),y))(1); // ==> (3*x^2+1)/(2*x^2)

In other words, a direct command that will return the inverse function of a function argument.

-Dale-

[Joe Horn]

(10-07-2017 04:20 PM)DrD Wrote:  Function Example:
y:=1/(sqrt(2*x-3));

Needed is a built in command to essentially do this:
simplify(solve(x = (1/(sqrt(2*y-3))),y))(1); // ==> (3*x^2+1)/(2*x^2)

In other words, a direct command that will return the inverse function of a function argument.

solve(y=1/(sqrt(2*x-3)),x) does the job (and even returns the result simplified if you have the "Simplify" CAS Setting on Maximum). Do you want something simpler than that?

[DrD]

I do want something simpler than that. There is a lot of use for inverse functions in math education. This, alone, should justify having a dedicated command that simply takes a function and returns it's inverse function. We have commands on the prime that are far less commonly used, and one command that seems to indicate that it WOULD be useful for inverse functions (according to the help docs), but isn't.

The solve command has many discussions in the forum, and has been frustrating. It takes additional presence of forethought to relate the idea of obtaining an inverse function, by using a solver.

So yes, a new command, perhaps invf(Func), would be useful, especially for students learning calculus. I would envision such a command to also return a simplified result. This would allow students to quickly confirm intermediate results and verify problem solution progress.

I could create a program for this, which would be contrary to one of the benefits of purchasing a powerful hand held calculator designed for students and educators. Ideally, as I encounter math problems, I want a device with an easy to understand, fully capable interface, that is up to the tasks required. That is where the value is for me, as a calculator customer. It's more than just the pride of ownership, I want the utility value.

-Dale-

[Joe Horn]

How do you teach a student to find the inverse function of y=f(x) by hand? Isn't it by solving that equation for x (that is, isolating x on one side of the equation)? And it teaches what an inverse function IS, and exercises the student's algebraic skills. After learning it that way, using SOLVE to find inverse functions would come naturally.

That seems far more educational than telling the student, "just press the InverseFunction key; it lets you get the answer without requiring you to understand what inverse functions are, or anything else for that matter."

Disclaimer: The above was my opinion when I wrote it, but might no longer be my opinion by time you read it. I reserve the right to keep learning and to adjust my opinions accordingly.

[DrD]

By that reasoning a student shouldn't use technology during the educational process. You could use similar rationale for advanced education and on into career and retirement. Thanks, just the same, but I prefer to use the prime for educational advantage. Somehow it seems obvious that the use of these implements makes education faster, easier, and better. Which is why an inverse function command is a good idea, in my opinion, which may also change, if I get a different opinion of that opinion.

[Joe Horn]

Just thinking out loud...

I see your point, and certainly agree on the educational value of technology, having taught high school math with HP calculators for decades. Perhaps my unspoken fear is the "slippery slope" of portable CAS systems, approaching the limit of voice recognition of every kind of problem and instant output of the solutions, requiring the user to know absolutely nothing. That would be a great tool in the field (!), but a TERRIBLE thing to teach with. No learning happens at all when the students have technology at hand which can do all their homework and take their tests for them, thus eliminating all incentive they may have to learn anything.

However, learning does happen in stages, so learning happens best when students have technology at hand which is optimized for learning exactly what it is that they are learning at that stage. Will somebody ever design a handheld which adapts itself to the student's current stage of math learning? Maybe. But nobody has done it yet.

[DrD]

Totally agree. I have heard that robotic teaching, with adaptive teaching methods tailored to each students individual learning rate, is already being developed. So technology is just getting started in educational fields.
I wonder if hp will be building any of that hardware?

[DrD]

Incidentally, where is the greater hurt? Typical conventionally educated person, that doesn't remember unused details after education, or technology based education, wherein one immediately gains useful solutions, and can derive value instantaneously, at any point in time?

[parisse]

(10-08-2017 04:18 AM)DrD Wrote:  Incidentally, where is the greater hurt? Typical conventionally educated person, that doesn't remember unused details after education, or technology based education, wherein one immediately gains useful solutions, and can derive value instantaneously, at any point in time?

There is a high risk of replacing teaching how to do it with your brain by teaching how to press a succession of keys, where the succession of keypress will not be valid anymore in a few years because UI will have changed. This can also happen with a builtin command because the name might change, and there is a good reason why there is no universal commandname for that:
1/ there is no magic way to invert any function (solve can not solve symbolically any equation)
2/ even if solve can solve the equation, there might be several solutions, and no way to choose one.
I can extend @@ (compose power) to negative exponents by inverting a function if solve can solve the equation (choosing the first solution), but I'm not convinced it will be better.

[DrD]

There is a tendency to move away from the idea of keypress technology. In time, other methods, including a biological interface, may become practical. Regardless of the technology in place as of a certain time, the idea of information processing using advanced technology is worthwhile.

Using a shopping cart, is better than carrying many items by hand, to the check out stand, possibly needing to go back for more. Hand carrying those items helps to assure ones memory where each product is within the store, but getting in and out of the store faster, is more efficient and results in greater sales volume.

There is a parallel in information technology. It's also inevitable at this point, isn't it? Students already have an option of stay at home (on line) education. That wasn't an option during my experience. Technology is pervasive in the classroom, including devices like the smartphone, and problem solving by example, is more visually and conceptually understandable, using multimedia. I don't see much argument against the idea. Keypress limitations, (as you describe) are but a mere moment on the scholastic timeline.

I agree that the solve command isn't ideal. In fact, the whole idea of calling inverse functions, inverses, is pretty lame. I mean if f(x) is a valid function, calling it's inverse f^-1(x), is just crazy, there is no mathematic recipropral in play there, (no exponent of -1, by the laws of exponents). That's what we call it, and in any event, it would be helpful to have a built in command to accommodate deriving so-called inverse functions.

[parisse]

I'm sorry but your arguments are too elaborate for my poor level in English.

[DrD]

I'm sorry for language issues. Technology is a good thing. It should be easy to use, and powerful. Using technology enables learning at a faster rate. Some American students, are using online courses as part of their regular classroom studies.

Who knows what tomorrow brings? It's ok if keypress technology goes away, something better will replace it. Some math teachers are terrible spellers, and some English teachers can not help solve math word problems! Both are well educated. Sometimes what was learned can be forgotten in time. Technology, in some form, will likely be able to help educators, students and others, recall forgotten details better.

Thank you for your help. I appreciate your help, and I hope others do also. Sometimes the prime can be difficult to use. My intention is to help identify areas where the prime seems like it could be improved. I use the prime quite often. Because I use it so often, I find problems where the prime isn't good. Often, it is my fault, and sometimes I'm pretty sure the prime could be improved.

I seek a built in inverse function command, that would be helpful for related problems. It should be an easy command to find in the toolbox, with understandable help documentation.

[Joe Horn]

There is nothing new under the sun.

[DrD]

Those keys do look like they have been under the sun, (perhaps too long)!

[jonmoore]

Apologies for dredging up this old discussion, but it's something I was teaching my youngest about last night so I thought it apt.

What the OP was asking for, is an equivalent of the DrInv command on the TI-86 (it's called DrawInv on the TI-84/83 family of calculators - ref: https://www.dummies.com/article/technolo...us-160952/)



In the classic "Investigations in Mathematics (for the HP 48G/GX)" a solution for the lack of this function on the 48G/GX was a clever program devised by Bill Wickes that drew a Parametric Plot over the original Function Plot.



People often ask why TI hammered HP in the education sector and the lack of this vital visualisation routine in HP calculators sums things up perfectly for me. Sure I understand Joe's pedagogic argument ref the by hand algebraic solution, but I've always found that using visualization to teach underlying algebraic rules is often the best way to cement algebraic insight.

The same authors behind Investigations in Mathematics (Donld's Latorre and Kreider) were also responsible for the massively influentic 'Calculus Conceps - An informal Approach' series of books. The content for this book was originally devised at Dartmore College (of BASIC fame) when the HP-28C was released in the late eighties. The book series started with the HP-48G/GX front and centre, alongside TI's 83/84/85 & 86 (it was first released in the early/mid nineties), but by the early naughties, the book series focus had moved to the TI's 84 & 89 alone (and Microsoft Excel).

There's no argument to the fact that HP's 48/49/50 series of calculators are more powerful and flexible than anything TI have ever released. But TI pumped lots of resources into curriculum outreach. Their calculators have less functions than HP calculators because it's what teachers and the majority of students wanted. HP may have released cheaper models for the education market with less features but the included functions were never as finely tuned to curriculum needs as those provided with the TI's.

Getting back to DrInv/DrawInv. The thing that amazes me, is that HP got feedback from LaTorre & Krieder back in the early nineties that this was an important function, and Bill Wickes devised a program to remidy that need. But that's as far as it went. The HP design team didn't use that feedback to provide a 'draw inverse function' capability into later calculators. In the preliminary edition of the graphing calculator instruction guide for 'Calculus Concepts - an Informal Approach', the HP 48G/GX required 27 bespoke routines to be entered into the calculator (some very lengthy), so it could be used to follow the text (the TI-83 and 85 only required 5 bespoke programs as everything else was already part of their native function-sets).

There were many other great routines for the 48 included in the Calculus Concepts graphing calculator guides as well as the draw inverse function routine. But none of these routines were included in future HP calculators. All of the bespoke programs for the TI 83/85 became native functions in future TI models (things like logistic regression and more flexible curve fitting). I'm not sure if it was oversight or arrogance that meant the HP design team didn't see the need (and benefit) of integrating any of the bespoke routines in future graphical models but it was most definitely a major oversight. TI cemented their reputation for providing curriculum focused solutions and HP increased their reputation for providing powerful solutions where complexity was often a barrier to usability - and this is the last thing you need in a classroom scenario, as you spend more time teaching the calculator feature-set, than the core mathematical concepts of the lesson content!

[Albert Chan]

(08-02-2022 12:22 PM)jonmoore Wrote:  What the OP was asking for, is an equivalent of the DrInv command on the TI-86

I think OP was asking more than a pretty picture, but actual inverse function.

Drawing it is easy, even without "DrInv" command.
Just plot -f(x), then turn the plot anti-clockwise 90°

[jonmoore]

(08-03-2022 04:19 PM)Albert Chan Wrote:  

(08-02-2022 12:22 PM)jonmoore Wrote:  What the OP was asking for, is an equivalent of the DrInv command on the TI-86

I think OP was asking more than a pretty picture, but actual inverse function.

Drawing it is easy, even without "DrInv" command.
Just plot -f(x), then turn the plot anti-clockwise 90°

It's so easy that it wasn't possible on the 48GX without a bespoke program that relied on parametric plotting!

The inverse of a function algebraicly ends is just the same. The graphical representation ends up plotting the y values on the x axis and the x values on the y axis. That's why it's such a useful insight for students studying precalculus.

Here's a Desmos example (and don't forget we're talking about high school students here).

https://www.desmos.com/calculator/oxmgrvftez

Things of course become more complicated as the complexity of the functions increases (complex functions obviously don't play nicely on the cartesian plane).

BTW it's not a -45 degree rotation, it's a reflection.

[Albert Chan]

(08-03-2022 05:26 PM)jonmoore Wrote:  

(08-03-2022 04:19 PM)Albert Chan Wrote:  Drawing it is easy, even without "DrInv" command.
Just plot -f(x), then turn the plot anti-clockwise 90°

BTW it's not a -45 degree rotation, it's a reflection.

That's what the sign, −f(x) is for.
Plots gives points (x,-y). Rotate calculator 90°, we have points (y,x)

[jonmoore]

(08-03-2022 06:20 PM)Albert Chan Wrote:  

(08-03-2022 05:26 PM)jonmoore Wrote:  BTW it's not a -45 degree rotation, it's a reflection.

That's what the sign, −f(x) is for.
Plots gives points (x,-y). Rotate calculator 90°, we have points (y,x)

But you're forgetting that the 48 series of calculators doesn't understand negated f(x) notation. e.g. If you input 'f(x)=2*x+3' and input the command DEF, this automatically gets translated to the RPL statement:

<< -> X '2*x+3' >>

However, if you input '-f(x)=2*x+3' and attempt to DEF that statement you get an invalid definition. This is the reason that Bill Wickes had to devise a workaround for Donald LaTorre's books.

My post dealt very specifically with the 48G series of calculators but the same invalid definition error applies to the 50g (and other RPL calculators) too.

Other graphing calculators may or may not work more gracefully out of the box with inverse notation but not HP's RPL calculators.