Defining Maximum and Minimum Functions for Common Domains

In summary, the maximum and minimum functions are defined as follows for any x belonging to a common domain D: max(f,g)(x) = f(x), if f(x) >= g(x)max(f,g)(x) = g(x), if g(x) > f(x)andmin(f, g)(x) = f(x), if f(x) <= g(x)min(f, g)(x) = g(x), if g(x) < f(x)To define the max(sinx, cosx) and min(x, x^2), we can use the piecewise definition or the absolute value method. The behavior of the functions on their respective intervals can be analyzed by finding points of equality and determining which
  • #1
irony of truth
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I want to let f and g be functions to be defined on a common domain D. Then the maximum and minimum functions are defined as follows for any x belonging to D...

max(f,g)(x) = f(x), if f(x) >= g(x)
max(f,g)(x) = g(x), if g(x) > f(x)

and

min(f, g)(x) = f(x), if f(x) <= g(x)
min(f, g)(x) = g(x), if g(x) < f(x)

How do I define my max(sinx, cosx) and min(x, x^2)?
 
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  • #2
think about it: when is cos greater than sin over the same domain? clearly the min and max functionals will depend upon the value of theta.

same for x and x^2 - clearly x is greater than x^2 for values less than 1, and less than x^2 for values greater than 1.
 
  • #3
irony of truth said:
I want to let f and g be functions to be defined on a common domain D. Then the maximum and minimum functions are defined as follows for any x belonging to D...

max(f,g)(x) = f(x), if f(x) >= g(x)
max(f,g)(x) = g(x), if g(x) > f(x)

and

min(f, g)(x) = f(x), if f(x) <= g(x)
min(f, g)(x) = g(x), if g(x) < f(x)

How do I define my max(sinx, cosx) and min(x, x^2)?
|f-g|=max(f,g)-min(f,g)
so
max(f,g)=(f+g+|f-g|)/2
min(f,g)=(f+g-|f-g|)/2
also the functions you are considering are continuous so if you find when they are equal, you can find which is greater on an interval where they are always not equal by considering any point in that interval.
for example
f=3x+2
g=2x-3
f=g->x=-5
so we consider
x<-5 then g>f
x>-5 then f>g
or we could try
max(f,g)=(3x+2+2x-3+|3x+2-(2x-3)|)/2
=(5x-1+|x+5|)/2
=(6x+4)/2=3x+2 x>-5
=(4x-6)/2=2x-3 x<-5
 
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  • #4
lurflurf said:
|f-g|=max(f,g)-min(f,g)
so
max(f,g)=(f+g+|f-g|)/2
min(f,g)=(f+g-|f-g|)/2
also the functions you are considering are continuous so if you find when they are equal, you can find which is greater on an interval where they are always not equal by considering any point in that interval.
for example
f=3x+2
g=2x-3
f=g->x=-5
so we consider
x<-5 then g>f
x>-5 then f>g
or we could try
max(f,g)=(3x+2+2x-3+|3x+2-(2x-3)|)/2
=(5x-1+|x+5|)/2
=(6x+4)/2=3x+2 x>-5
=(4x-6)/2=2x-3 x<-5

I want to know how you got |f-g|=max(f,g)-min(f,g)...

Also, will that example you posted be my guide to solve this? So, I have to say the interval in which the two functions are as designated...
 
  • #5
quetzalcoatl9,

You mentioned that "think about it: when is cos greater than sin over the same domain? clearly the min and max functionals will depend upon the value of theta"

The graph seems to look like "lots of McDonald's M"

You also mentioned that "same for x and x^2 - clearly x is greater than x^2 for values less than 1, and less than x^2 for values greater than 1. "

x > x^2 when 0 < x < 1... not less that 1.. :D
x < x^2 when x > 1 or x < 0.

x = x^2 when x = 0 or 1.
 
  • #6
irony of truth said:
Also, will that example you posted be my guide to solve this? So, I have to say the interval in which the two functions are as designated...

Sine and Cosine are uniquely defined by their behavior on [0, 2pi ). So if you define the behavior of max(Sin(x), Cos(x)) on that interval, by extension you know what it looks like everywhere.
 
  • #7
irony of truth said:
I want to know how you got |f-g|=max(f,g)-min(f,g)...

Also, will that example you posted be my guide to solve this? So, I have to say the interval in which the two functions are as designated...

max(f,g)=min(f,g)->f=g
max(f,g)>=min(f,g)
so max(f,g)-min(f,g)>=0
and either max(f,g)-min(f,g)=f-g or max(f,g)-min(f,g)=g-f
hence
max(f,g)-min(f,g)=|f-g|
There are two easy ways to do these.
They are easy because the functions are continuous and because they are equal only on discrete sets. That is for every x such that f(x)=g(x) there exist a positive number h so that x is the only point of equality in (x-h,x+h).
Method 1: piecewise definition
here we aim to partition the real numbers into intervals and specify which function to use on the interval. ie
x<a
a<x<b
b<x<c
c<x
with f(x)=g(x) if x=a,b,c
so on any interval where f and g are continuous and never equal |f-g| never changes sign thus if f>g for any point in such an inteval f>g for all points.
method 2: absolute value
max(f,g)=(f+g+|f-g|)/2
min(f,g)=(f+g-|f-g|)/2
allow us to right out the min and max right away
sometimes further simplification is possible
sometimes we use this to give the min and max as piecewise defined functions
 
  • #8
Hmm, thank you for the help...

If I were to define my function, can I just "plugged in" the f(x) and g(x) with their corresponding functions.. then just analyze the behavior of their graphs, of course, I will have to find those intervals?

max(sinx, cosx)

max(f,g)(x) = sinx, if sinx >= cosx
max(f,g)(x) = cosx, if cosx > sinx
 
  • #9
irony of truth said:
Hmm, thank you for the help...

If I were to define my function, can I just "plugged in" the f(x) and g(x) with their corresponding functions.. then just analyze the behavior of their graphs, of course, I will have to find those intervals?

max(sinx, cosx)

max(f,g)(x) = sinx, if sinx >= cosx
max(f,g)(x) = cosx, if cosx > sinx
find out when sin(x)=cos(x)
then show sin(x)-cos(x) changes sigh at all of its zeros
then which function (sin(x) or cos(x)) is used in each interval can be found by finding which is used for any point in the interval
as has been said you can do any interval of length 2pi and then extend to all numbers.
max(f,g)(x) = sinx, if sinx >= cosx
max(f,g)(x) = cosx, if cosx > sinx
is a fine definition
as is
max(f,g)(x)=(sin(x)+cos(x)+|sin(x)-cos(x)|)/2
but you probably what to express the conditions
sin(x) > cos(x)
cos(x) > sin(x)
in a more useful form
what properties of max(sin(x),cos(x)) do you want to highlight?
 
  • #10
Ok... I got it... thank you for the help
 

FAQ: Defining Maximum and Minimum Functions for Common Domains

What is the purpose of maximum and minimum functions in scientific research?

The purpose of maximum and minimum functions is to help identify the highest and lowest values in a set of data. This can be useful in various scientific fields, such as statistics, economics, and engineering, to determine the most extreme data points and make decisions based on them.

How do maximum and minimum functions work?

Maximum and minimum functions work by comparing a set of data and returning the highest and lowest values. In mathematical terms, the maximum function returns the largest value, while the minimum function returns the smallest value. This is typically done through algorithms or equations that analyze the data and determine the extreme values.

Can maximum and minimum functions be used with any type of data?

Yes, maximum and minimum functions can be used with numerical, categorical, and even binary data. The function will simply compare the data and return the highest or lowest value, regardless of the type of data. However, it is important to note that the data must be comparable for the function to work accurately.

Are maximum and minimum functions important in data analysis?

Yes, maximum and minimum functions are crucial in data analysis as they help identify outliers and extreme data points that may significantly affect the results. They can also be used to determine trends and patterns in the data, which can be valuable for making informed decisions and drawing conclusions.

Are there any potential limitations of using maximum and minimum functions?

One potential limitation of using maximum and minimum functions is that they only consider the highest and lowest values and do not take into account the entire dataset. This can result in misleading conclusions if the data is not properly analyzed or if there are other important factors to consider. Additionally, extreme values may not always be accurate or representative of the overall data, so caution should be taken when relying solely on these functions.

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