# What is the best function of the following shape?

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M on 6 Sep 2022
Edited: Bruno Luong on 12 Sep 2022
How can I construct this shape? what is the best function of it?
It is more like sigmoid function + straight line
But I cant get how can I represent it by one equation and How to control its limits!
The Y limits should be from 0 to 1
( I want to use it as a fuzzy memebership function)
Is there any recommendation?
Sam Chak on 7 Sep 2022
Not exactly sure what you want to do because you didn't define the problem in a mathematical manner. The existing functions in the Fuzzy Toolbox library stock do not have cutoff points. Guess you want to create a custom Sigmoidal Membership Function with one-sided cutoff point for your unique Fuzzy System, where it can be rescaled and reflected in anyway you want:
M on 7 Sep 2022
Hi @Sam Chak, How did you get the above plots? Could you please provide me with the code.
I want to use this membership function in a code, not with the fuzzy tool box. Thanks

Sam Chak on 7 Sep 2022
Edited: Sam Chak on 12 Sep 2022
Hi @M
The original code was accidentally highlighted and deleted due to sensitive touchpad. It was constructed based on @Torsten's idea on the Arctangent function but I replaced it with the Logistic function as shown in this link:
Since you don't want anything to do with the Fuzzy Logic Toolbox, this new solution, a Quintic function (not in the toolbox, so you can probably claim novelty in the Fuzzy MF paper), is tailor-made for you so that you adjust the position of the cutoff point (end point of blue line) easily.
Example #1: The Quintic function, a 5th-order polynomial, is commonly used in the trajectory planning of a mobile robot. So I modified the function for a custom fuzzy membership function.
I have not tested everything. But this is the basic concept for you to develop. Hope it helps.
% Quintic function-based MF (blue line as main)
% Start point (xA, yA); End point (xB, yB);
xf = 1;
x = linspace(0, xf, xf*1e4 + 1);
yA = 0;
yB = 1;
xA = 0.1;
xB = 0.9;
y = (yA + (10*((x - xA)/(xB - xA)).^3 - 15*((x - xA)/(xB - xA)).^4 + 6*((x - xA)/(xB - xA)).^5)*(yB - yA)).*((0 <= (x - xA)) & ((x - xA) < (xB - xA)));
plot(x, y, -(x-xf), y), grid on, ylim([0 1.2])
legend('MF_1', 'MF_2', 'location', 'north')
xlabel('Universe of Discourse, \it{x}'), ylabel('\mu(\it{x})')
Example #2: Using the Logistic function:
xf = 1;
x = linspace(0, xf, xf*1e4 + 1);
xA = 0.1; % Left bound of the blue S-curve
xB = 0.9; % Right bound of the blue S-curve
a = 2*9.2; % (logistic growth rate) adjust the steepness of S-curve
c = 0.5; % shift the center of S-curve to x = c
y = sigmf(x, [a c]).*((0 <= (x - xA)) & ((x - xA) < (xB - xA)));
plot(x, y, -(x-xf), y), grid on, ylim([0 1.2])
legend('MF_1', 'MF_2', 'location', 'north')
xlabel('Universe of Discourse, \it{x}'), ylabel('\mu(\it{x})')
Sam Chak on 12 Sep 2022
@M, Theoretically, the polynomial function offers more flexibility in adjusting the shape of the S-curve in anyway you like as long as it is monotonically increasing/decreasing, and if you know how to adjust the coefficients of the terms.
For simplicity, you may want to choose the Logistic function (aka sigmf), because you just need adjust one parameter, the logistic growth rate (a) to shape the steepness of the S-curve.
Bruno Luong on 12 Sep 2022
((0 <= (x - xA)) & ((x - xA) < (xB - xA)))
or equivalently
((x >= xA) & (x < xB))

Torsten on 6 Sep 2022
Edited: Torsten on 6 Sep 2022
x = -2:0.01:2;
plot([x,x(end)],[1/pi*atan(2*x)+1/2,1/pi*atan(2*x(1))+1/2])
xlim ([-3 3])
ylim ([0 1])
M on 7 Sep 2022
Ok @Torsten, Thanks
M on 7 Sep 2022

Bruno Luong on 7 Sep 2022
Try this:
xmin = 1;
xmax = 3;
pp = struct('form','pp',...
'breaks',[-1e-100 0 1 inf],...
'coefs',[0 0 0 0 0 0;6,-15,10,0,0,0;0 0 0 0 0 1],...
'pieces',3,...
'order',6,...
'dim',1);
sfun = @(x) ppval(pp,(x-xmin)/(xmax-xmin));
ezplot(sfun,xmin-1,xmax+1)
Walter Roberson on 12 Sep 2022
In analysis, a "function" is any mapping of values to other values. It might or might not be "one to one" or "onto". There is no requirement that the function maps all possible inputs in the domain.
In analysis, functions over infinite sets are treated like the limit of finite mappings. For example, x->x^2, x in Z, is treated as-if it were the set of mappings {..., -2 -> 4, -1 -> 1, 0 -> 0, 1 -> 1, 2 -> 2, ...} rather than as being something fundamentally different. There does not have to be a formula for a function: the complete list of mappings defines the function; formulae can make it significantly more compact to express though. The mappings are the fundamental property of the function; formulae are conveniences to express mappings.
Bruno Luong on 12 Sep 2022
Edited: Bruno Luong on 12 Sep 2022
"There is no requirement that the function maps all possible inputs in the domain."
Wrong to my book, all elements of input set (domain) must be mapped (defined) by the function.

Walter Roberson on 6 Sep 2022
Looks like tansig
M on 6 Sep 2022
Edited: M on 6 Sep 2022
@Walter Roberson, I need a general eqaution for this shape so I can adjust the hight and width of it depending on my application ( I want to use it as a fuzzy memebership function). Note that there is a left straight line to the curve, what could be the best eqaution that represents shape?
Image Analyst on 12 Sep 2022
@M there are a variety of equations that can do that. See
and just pick one.

James Tursa on 6 Sep 2022
Edited: James Tursa on 6 Sep 2022
There are lots of functions that could fit this general shape. E.g., sin(x), or more generally A*sin(B*x+C) depending on width and height of the graph which you don't show. Maybe you could share some more details about where this shape comes from and the height and width sizes.
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M on 6 Sep 2022
Edited: M on 6 Sep 2022
@James Tursa, I think it could be sigmoid function + straight line
But cant get how can I represent it by one equation and How to control its limits!
Image Analyst on 12 Sep 2022
@M there are a variety of equations that can do that. See
and just pick one.