# Function Composition of Multivariate Functions

• Prof. 27
In summary, the conversation is about a student struggling with a homework problem involving multivariate functions and function composition. They mention finding an extension of composition on Wikipedia but have not learned it in class. The problem involves calculating T ◦ S, writing the matrices of T and S, and finding the matrix of T ◦ S. The student is unsure about the order of the variables in the function compositions and how it affects the final result. They seek clarification and help.
Prof. 27

## Homework Statement

This is a homework problem for my Honors Calculus I class. The problem I'm having is that though I can solve a traditional function composition problem, I'm stumped as to how to do this for multivariate functions. I read that it requires an extension of the notion of composition itself on Wikipedia (to partial composition I believe it says). Yet we have not learned this, nor is it anywhere to be found in the lecture notes, which until this point have been excellent. My question is, is there something I'm not understanding? Is this possible with traditional composition? I'll show the idea I had but I don't know if it is anywhere near correct. Bold parts are not currently relevant to what I don't understand.

Question:

Let T : R2 → R3 be T(x, y) = (2x−3y, 4x−11y, x), and let S : R3 → R2 be S(u, v, w) = (u + v − w, 2u − 3v + 5w). i) Calculate T ◦ S; ii) Write the matrices of T and S; iii) Find the matrix of T ◦ S and relate it to the matrices in ii).

My Attempt:
We know at least that ToS must be in R5, because the Cartesian product of two sets of finite dimensions equals the dimensions added.

T(x,y) = 2x-3y, 4x-11y, x
S(u,v,z) = u+v-w, 2u-3v+5w

a = u+v-w
b = 2u-3v+5w

T(S(u,v,z) = T(a,b) = 2a-3b, 4a-11b, a

The problem: Isn't assigning a,b to x,y respectively arbitrary? Couldn't I just as well say,

T(S(u,v,z) = T(b,a) = 2b-3a, 4b-11a, b
Since a does not equal b, T(b,a) does not equal T(a,b) (at least usually).

Help?

None

## The Attempt at a Solution

Stack Exchange
Lecture Notes
Textbook, Calculus by Michael Spivak

Prof. 27 said:

## Homework Statement

This is a homework problem for my Honors Calculus I class. The problem I'm having is that though I can solve a traditional function composition problem, I'm stumped as to how to do this for multivariate functions. I read that it requires an extension of the notion of composition itself on Wikipedia (to partial composition I believe it says). Yet we have not learned this, nor is it anywhere to be found in the lecture notes, which until this point have been excellent. My question is, is there something I'm not understanding? Is this possible with traditional composition? I'll show the idea I had but I don't know if it is anywhere near correct. Bold parts are not currently relevant to what I don't understand.

Question:

Let T : R2 → R3 be T(x, y) = (2x−3y, 4x−11y, x), and let S : R3 → R2 be S(u, v, w) = (u + v − w, 2u − 3v + 5w). i) Calculate T ◦ S; ii) Write the matrices of T and S; iii) Find the matrix of T ◦ S and relate it to the matrices in ii).

My Attempt:
We know at least that ToS must be in R5, because the Cartesian product of two sets of finite dimensions equals the dimensions added.

T(x,y) = 2x-3y, 4x-11y, x
S(u,v,z) = u+v-w, 2u-3v+5w

a = u+v-w
b = 2u-3v+5w

T(S(u,v,z) = T(a,b) = 2a-3b, 4a-11b, a

The problem: Isn't assigning a,b to x,y respectively arbitrary? Couldn't I just as well say,

T(S(u,v,z) = T(b,a) = 2b-3a, 4b-11a, b
Since a does not equal b, T(b,a) does not equal T(a,b) (at least usually).

Help?

None

## The Attempt at a Solution

Stack Exchange
Lecture Notes
Textbook, Calculus by Michael Spivak
When using the terminology:: "Ordered Pair", and/or "Ordered Triple";

the word order is very important.

Thanks, that helps a lot.

## 1. How is function composition defined for multivariate functions?

Function composition of multivariate functions is the process of combining two or more functions to create a new function. In other words, the output of one function becomes the input of another function. This can be represented as (f ∘ g)(x) = f(g(x)), where f and g are functions and x is the input variable.

## 2. What is the purpose of using function composition for multivariate functions?

The purpose of function composition is to break down complex functions into smaller, more manageable components. It allows us to apply multiple functions to a single input, making it easier to solve problems involving multivariate functions.

## 3. How do you determine the domain and range of a composed multivariate function?

The domain of a composed multivariate function is the set of all possible input values that can be used for the innermost function. The range, on the other hand, is the set of all possible output values that can be obtained from the outermost function. It is important to note that the domain and range of a composed function may be limited by the domains and ranges of the individual functions involved.

## 4. Can any two multivariate functions be composed together?

No, not all multivariate functions can be composed together. In order for function composition to be possible, the output of the inner function must match the input of the outer function. This means that the domains and ranges of the individual functions must be compatible.

## 5. How is the order of functions important in function composition of multivariate functions?

The order of functions is crucial in function composition because it determines the final result of the composed function. Changing the order of functions can lead to different outputs, and in some cases, the composed function may not even be valid. Therefore, it is important to carefully consider the order of functions when composing multivariate functions.

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