Function challenge?

Question

think of a function as a math machine with an input an output Suppose the function A(x)=3x+1 that means if u put any number (x) into this function machin the machin will multiply the number by 3 and add 1 for example what is the output if 5 is the input? 5 times 3 plus 1=16 for the output thats written A(5)=16 suppose u join 2 function machines so that the output of the 1st one is connected to the input of the 2nd one lets make the 2nd function B(x)=x^2 if u put 2 into the A machine out comes 7 then seven is the input to the B machine which squares it and out comes 49 thats written B(A(2))=49
here are 5 functions
A(x)=2x-3
B(x)=x^2
C(x)=2x+1
D(x)=4x^2
E(x)=7x+5
i am going to connect them in alphabetical order and use 2 as the input number that would be written as E(D(C(B(A(2))))) the resulting output = ?
what arrangement of functions will produce the largest output with 2 as the input (each function used only once) i am looking for logical reasoning and algebra. Explain?

Answer 1

B(D(A(E(C(2))))) = 562,448,656
You have two sets of functions, degree one and degree two.
The degree one function which will product the greatest value with a given positive input is E(x) = 7x+5.
Of the degree two equations, 4x^2 > x^2 for all positive x,
and x^2 > 7x+5 for integers greater than 7.
Therefore the degree two functions should be last.
Since 4(x^2)^2 < (4x^2)^2, B should be last, D should be next.
Of the six combinations of C, E, A (which come first), the one with the greatest output is A(E(C(X))) = 28x + 21.

E(A(C(X))) = 28X - 2 ; A(C(E(X))) = 28x + 19 ;
E(C(A(X))) = 28X - 30 ; C(A(E(X))) = 28X + 15;
C(E(A(X))) = 28X - 31

Answer 2

What you're talking about is called 'composition of functions' or a 'composite function' and they're fairly important in higher math (especially modern algebra).

It's fairly straightforward to calculate what the composite function will do (increase, decrease) in the neighborhood of a given argument as the value of the argument increases or decreases (assuming that all of the functions are continuous) but, unfortunately, there really isn't any good way to predict what the final value of a composit function will be except to calculate it for the given argument.

Since all of the functions you've written down (except A) return larger values than the argument for positive arguments, then you probably want A as the last (outermost) function.

Since 14x+12 > 14x+11 I'd go with E(C(2))

Since 16x^4 > 4x^4 B(D(E(C(2)))) makes sense so I'd guess that
A(B(D(E(C(2))))) = 81,919,997 would be the largest value.for 2. But it might well be a different composition for an argument < 1 or negative.


Doug

Answer 3

Congratulations! You explained very well the proposition.
Unfortunately, there is no rule in this case.
You can get the answer by trying:
D and B must be the last...
a) first you try ACE,AEC,CAE,CEA,EAC and ECA and get the great value.
b) second you try DB and BD with the great value.