S-curve and second power polynominal curve - need to combined into one equation

[Bob52]

Thank you in advance

?=[2∗(?/?)^2−(?/?)^4]∗? from 0 to f
y=[((1−?)(?−?)^2))/(ℎ−?))^2]+? from f to h

The first equation is an s-curve that starts at (0,0) and goes to (f,g) The second curve starts the same point (f,g) and goes to (h,1), therefore, this is a scale factor equation. How do you combine these two equations into a single equation that goes from (0,0) to (h,1). Please show the steps.
I have tried to differentiate each equation. Add those together and then integrate over the given limits. It does not work. Please help.
Once again, I apreciate your efforts.

 

[Dr.Peterson]

Why do you think this can be done (in a reasonable and useful way)? Why do you want to do it? Why do you think differentiating and adding would accomplish your goal, when the two parts are defined over different domains?

What you have is a piecewise-defined function. That is often the most useful form for a function; there is no benefit in trying to write one equation for the whole domain. It is just as valid a function whether there are one equation or two.

The best thing you can do is to tell us the context of this question, so we can suggest a better way to approach your real goal.

 

[Bob52]

Wow, that is the correct answer! What are your steps? I started with the derivative of each, getting 2x and 4x^3, respectfully. Then I added those together, 2x+4x^3 and integrated from 0 to 2 and got 2x+4x^3, which is the wrong answer. Please help with the steps. Thanks again.

 

[Dr.Peterson]

Wow, that is the correct answer! What are your steps? I started with the derivative of each, getting 2x and 4x^3, respectfully. Then I added those together, 2x+4x^3 and integrated from 0 to 2 and got 2x+4x^3, which is the wrong answer. Please help with the steps. Thanks again.

I'm not sure what you are saying is the correct answer. Please answer my questions.

As far as I am concerned, what you originally wrote, restated as a piecewise function, is the correct "answer":

[MATH]y=\left\{\begin{matrix} \left[2\left(\frac{x}{f}\right)^2-\left(\frac{x}{f}\right)^4 \right]\cdot g & \text{if }0\le x < f\\ \left(\frac{(1-g)(x-f)^2}{h-f} \right)^2+g & \text{if }f\le x < h \end{matrix}\right.[/MATH]
There is absolutely no reason to do any of what you did; the two parts of the function do not exist on the same domain, so they can't be added.

 

[JeffM]

I'm not sure what you are saying is the correct answer.

I'm not sure we know is the correct question.

Once again, we ask to have the complete, exact wording of the problem.

 

[Bob52]

Attached is a PDF with explanation. Thanks

 

[JeffM]

Most of the functions that you meet in algebra or calculus can be described by a single formula. But that is NOT AT ALL a requirement for a function.

The function you are talking about is described by two different formulas. That is what is meant by a function defined piecewise: different rules for different pieces. To describe this function takes two different formulas.

There is no way to splice them together into a single formula.

 

[Deleted member 4993]

I'm not sure we know is the correct question.

Once again, we ask to have the complete, exact wording of the problem.

I think S/he switched threads. I think S/he was responding to this thread:

Create One Equation out of two Equations

Thank you in advance. I do not know how to combine two continuous definitive equations (using derivatives and definitive integrals?) into one equation. They have identical tangents at their intersection. Equation One: y=(x-squared) from 0 to 1 Equation Two: y=(x-to the fourth power)...

www.freemathhelp.com

 

[JeffM]

I think S/he switched threads. I think S/he was responding to this thread:

Create One Equation out of two Equations

Thank you in advance. I do not know how to combine two continuous definitive equations (using derivatives and definitive integrals?) into one equation. They have identical tangents at their intersection. Equation One: y=(x-squared) from 0 to 1 Equation Two: y=(x-to the fourth power)...

www.freemathhelp.com

Thanks, O Khan of Khans.

I suspect that our student may simply believe that a function must have a single formula over its entire domain.

 

[Bob52]

 

[Dr.Peterson]

[copied from PDF]
Thanks for your input! Somehow my original equations were transposed improperly,
so I have rewritten them below in a better format. I also added a graph of them.
I do not understand when you say they are not in the same "domain". As
mentioned above, I have tried and tried to write one equation from 0 to h.
I have triple checked my initial equations and they are correct. Obviously, if no
equation exists then I will understand. Thanks again. You guys are amazing.

Here is my version, corrected:
[MATH]y=\left\{\begin{matrix} \left[2\left(\frac{x}{f}\right)^2-\left(\frac{x}{f}\right)^4 \right]\cdot g & \text{if }0\le x < f\\ \frac{(1-g)(x-f)^2}{(h-f)^2}+g & \text{if }f\le x < h \end{matrix}\right.[/MATH]
In fact, I had previously made a graph for the case where f and g are 1/2 and h is 1, correctly guessing what you really meant:


Your work in obtaining the equations you gave us is very good.

What we really want to know is why you want to get a single equation.

The domain of a function is the set of input values (x). The first part (my red) applies only when x is from 0 to f; the second part (my green) applies only when x is from f to h. These are two different functions with different domains -- they don't apply to the same values of x. So you can't add them and get something meaningful. (If you are learning calculus, this should be familiar to you.)

But as has been mentioned, in posts #2 and #7, there is no value in trying to combine these into one equation. Together, in the piecewise form I've shown, they are one function represented by two equations for different values of x, and that is a perfectly acceptable form for a function. It is also the most useful form almost always.

We want to know what your ultimate goal is. What is this for, and why do you think you need one equation? We can probably help you find a way to use what you already have to accomplish your goal.

 

[JeffM]

With great respect for Dr. Peterson, I believe the terminology of the last sentence of post 4 may be a tiny bit confusing. The domain of f(x) is the set of values of x for which f(x) is defined. In this case, the function is well defined for all non-negative values of x. In that sense, the function has one domain.

This is kind of nit picking because, in this and many other cases, no one formula applies to the entire domain. Instead, different formulas apply to different parts of the domain, which I would probably call sub-domains rather than different domains. When this occurs, it is called a piecewise definition as Dr. Peterson explained in post 2. In such cases, there is no need to give a single formula for the domain as a whole because the formulas for each piece give all the information required. Moreover, it may be (in fact usually is) impossible to reduce the different formulas to one consistent formula. Even if it were possible to do so, the resulting formula might be incredibly complex and very hard to understand.

So, again as Dr. Peterson, implied in post 2, there is nothing wrong at all in having different formulas for different sub-domains.