# Get envelope of a set of functions?

I'm new with mathematica and I'm trying to solve a problem. I have a set of functions in the same variable. Let's say the domain is [0,2]. I would like to get an ordered list of the functions representing the "envelope". I'll try to explain, I have these 4 linear functions red, orange, green, blue.

And this is the plot of the maximum among all the functions.

How can I get a list {red, orange, green, blue} saying which is the max function in each segment?

Thanks

• Max[f1, f2, f3, f4]? Apr 15, 2016 at 10:14

Michael E2 is correct in that Max is remarkably capable with functions as well as values.

Defining

f1[x_] := -2 x + 2
f2[x_] := -x + 1.5
f3[x_] := x - 0.5
f4[x_] := 2 x - 2

funcs = {f1[x], f2[x], f3[x], f4[x]};


We can use Max to get the function of the envelope:

m[x_] = PiecewiseExpand@Max[f1[x], f2[x], f3[x], f4[x]]


Plot[funcs, {x, 0, 2}, Epilog -> First@Plot[m[x], {x, 0, 2}, PlotStyle -> Directive[Dashed, Black]]]


Provided your functions are all linear we can safely solve for all the intersections and then examine which function is the largest in each region:

crossings = Union@N@Flatten[Outer[Solve[#1 == #2, x] &, funcs, funcs]][[;;, 2]];

Plot[funcs, {x, 0, 2}, Epilog -> {Line[{{#, -2}, {#, 2}}] & /@ crossings}]


Thus the different regions are bounded by:

regions =  Join[{x < crossings[[1]]}, (#[[1]] < x < #[[2]]) & /@ Partition[crossings, 2, 1], {crossings[[-1]] < x}]


{x < 0.5, 0.5 < x < 0.833333, 0.833333 < x < 1., 1. < x < 1.16667, 1.16667 < x < 1.5, 1.5 < x}

Within which the orderings are:

Reverse /@ (Ordering@Through[{f1, f2, f3, f4}[#]] & /@ MovingAverage[
Join[{crossings[[1]] - 1}, crossings, {crossings[[-1]] + 1}], 2])


{{1, 2, 3, 4}, {2, 1, 3, 4}, {2, 3, 1, 4}, {3, 2, 4, 1}, {3, 4, 2, 1}, {4, 3, 2, 1}}

• Thank you! Is there a way to substitute "True" with the real interval of x (that is 1.<x<=1.5) in the output of PiecewiseExpand? Apr 19, 2016 at 16:25

Let

f[x_] := {-x+2,-2x+3,1.1x-1,2x-3}
A=0;
B=3;
Plot[Evaluate@f[x],{x,A,B}]


With this,

Flatten[Ordering[#,-1]&/@f/@ Mean/@Partition[Sort@Flatten[{A,B,x/.Table[NSolve[{f[x][[i]]==f[x][[j]],A<=x<=B},x],{i,1,Length[f[0]]-1},{j,i+1,Length[f[0]]}]}],2,1]]//.{a___,b_,b_,c___}:>{a,b,c}


Outputs {2,1,3,4}. This code works also for non-linear functions.

For example, if

f[x_] := {Sin[x],-2x^2+3,1.1x^3-4,2x-3,4Cos[7x]}


the output is {5, 2, 5, 1, 5, 3}.