Questions tagged [function-and-relation-composition]
For questions about the composition of functions and relations.
945
questions
0
votes
1answer
26 views
Translation of the function in Complex
Consider the $T : z \to z+a$ for $a \in \mathbb{C}$ (Here the $a = \alpha + i \beta$)
Take any complex function, $f(z)$
Then, $T \circ f(z) = f(z) + a$ is translation by $a$ (I.e. $f(z)$ is translated ...
1
vote
2answers
47 views
Let 𝑎, 𝑏 ∈ ℝ and let 𝑓$_{𝑎,𝑏}$: ℝ → ℝ be the function defined as 𝑓$_{𝑎,𝑏}$ (𝑥) = 𝑎𝑥 + 𝑏 for all 𝑥 ∈ ℝ. Use this for the problems below.
a) Prove that 𝐺 = {𝑓$_{𝑎,𝑏}$ | 𝑎, 𝑏 ∈ ℝ, 𝑎𝑛𝑑 𝑎 ≠ 0} is a group, where the operation is composition.
Let 𝐺1 = {$\big(\begin{smallmatrix}
a & 0\\
b & 1
\end{smallmatrix}\big)$ ∶ �...
0
votes
0answers
23 views
Confusion regarding symmetric and anti-symmetric relation
let R
be a relation on a collection of sets defined as follows,
R
{
(
A
,
B
)
|
A
⊆
B
}
Here, since it is an improper set, can't we take it as A=B and say that it is both symmetric and anti-symmetric ...
0
votes
3answers
50 views
How can I prove that a function is invertible, where do I get the function and the necessary data?
"If g is another invertible function, then the compound function $f ∘ g$ is also
invertible, and it is fulfilled.."
$$( f ∘g) ^{-1} = g^{-1} ∘ f^{-1}$$
I don't even understand where to start....
0
votes
1answer
19 views
How do I prove or disprove these relation compositions? [closed]
So I am not told if these statements are true or not (although they appear true). I am asked to either prove them in general terms, or disprove them with specific examples. I am confused about ...
0
votes
1answer
16 views
Composition of a Realtion with identity is the relation itself
I am completely lost as to how to prove the following, although it appears straightforward. I know how this holds for functions, but how do I prove it specifically for relations? I need to provide a ...
-1
votes
0answers
29 views
Continuity possible?
Can functions $f$ and $g$ not be continuous at $8$ while function $f$ of $g$ is continuous at $8$?
So far I've tried using piecewise functions to accomplish this but haven't been successful.
1
vote
1answer
17 views
How to calculate the maximal domain and hence range in this question?
Question:
The function $f : R → R, f (x)$ is a polynomial function of degree 4.
Part of the graph of $f$ is shown below. The graph of $f$ touches the
x-axis at the origin.
part a) wants me to find ...
1
vote
3answers
79 views
Let $f(x) = |x+1|-|x-1|$, find $f \circ f\circ f\circ f … \circ f(x)$ (n times).
Let $f(x) = |x+1|-|x-1|$, find $f \circ f\circ f\circ f ... \circ f(x)$ (n times). I don't know where to start... Should I use mathematical induction? But what should be my hypothesis? Should I ...
1
vote
1answer
23 views
Calculating Composition of 2 Functions
Let $f: \mathbb{R}^2 \to \mathbb{R} $ and $g: \mathbb{R} \to \mathbb{R}^2$.
Suppose $g(0) = (1,2)$, $g'(0) = (-1,4)$ and $(f \circ g)'(0) = -3 $.
Furthermore $\displaystyle \frac{df}{dx}(1,2) = \frac{...
1
vote
0answers
54 views
Functional equation involving composition and exponential
My main question is how to solve this functional equation:
$f(f(x)) = e^x$.
The context within which this came up was when I was attempting to extend the notation of the inverse function ($f^{-1}$) to ...
0
votes
1answer
16 views
Finding the domain of a composition of functions where one function is given in terms of y and the other in terms of x.
For a math assignment I was assigned by my professor, I've been asked to find the domain of a composition of the following functions.
$f(y) =\frac{4}{y - 2}$
$g(x) =\frac{5}{3x - 1}$
I know that the ...
0
votes
1answer
30 views
Composite functions with more complicated functions
I have this question, as below, and I cannot understand what I am being asked of. Typically I would consider the composition of functions to be rather simple, such as $f(x)=2x-5$ $g(x)=x-2$, therefore ...
1
vote
1answer
26 views
How to See if a Composite Function is One-To-One and Onto
So we have $f$ that is onto, and $g$ which is onto and one-to-one. Is $f \circ g$ onto and one-to-one?
My attempt at to solve this problem was to state the following scenarios and come to a conclusion....
1
vote
1answer
32 views
Prove that any map $f : X \to Y$ can be represented as composition of 1) inj and surj. 2) surj. and inj.
I was given the following task
Prove that any map $f : X \to Y$ can be represented as
composition $f = g \circ h$, where $g$ is a surjection and $h$ is injection
composition $f = z \circ c$, where $...
0
votes
3answers
75 views
Why $f(g(x)) = x$ and $f(x)=g(x)$ imply $f(x)=x$?
I was watching this video from blackpenredpen where he solves the equation $\sqrt{5-x}=5-x^2$ by writing it in terms of "5". However, there's a comment with an alternate solution using ...
3
votes
1answer
69 views
How to determine the nature of two functions from their composite function?
The following question was asked in an examination:
Let $f$ and $g$ be two functions with domain and codomain equal to the set of real numbers. If,
$$g\circ f(x) =
\begin{cases}
x^2, & \text{if $...
1
vote
1answer
23 views
Correct Notation for Limits of Function Composition
When taking the limit of function composition, what is the best way to write the intermediate step?
Suppose we have two functions $f(x)$ and $g(x)$ that are continuous where $\lim_{x\to a} g(x)=b$ and ...
0
votes
1answer
37 views
Proving this continuous function is entire if a related function is given entire
This question was asked in a masters exam for which I am preparing and I was unable to think about it. So, I am asking it here.
Question: Let f be a continuous function from $\mathbb{C} \to \mathbb{C}...
1
vote
1answer
38 views
How is it possible for the composite function to be continuous in this case?
When
$f(x) = x^2+3x+1 (x\geq1), x+2(x<1)$,
$g(t) = ㅣ3t-12ㅣ$,
$t=f(x)$
In this case, how can $g(f(x))$ be define between when $f(x)= [3,5]$?
$f(x)$ do not have $3<f(x)\leq5$ as a result.
so $g(t)$...
0
votes
4answers
76 views
If $f(x)=\frac{1-x}{1+x}$, then how do I explain graphically why $f(f(x))=x$?
Let $f(x)=\dfrac{1-x}{1+x}$. Then $f(f(x))=x$.
The domain of $f(f(x))$ is $\mathbb{R}$\ ${-1}$
How do I explain graphically why the formula for $f(f(x))$ is the way it is?
0
votes
2answers
38 views
Odd and even properties of functions
Could I get a hint or a clue on how to solve this problem?
Problem:
($f$ is even if $f(-x)=f(x)$ and $f$ is odd if $f(-x)=-f(x)$.)
Suppose $f$ is a function defined on all of $\mathbb{R}$
(a)
Check ...
1
vote
2answers
56 views
Is $x-y \ne 0$ a transitive relation?
I know that for a relation R is transitive if, for all elements aRb and bRc implies aRc.
I came across this question, " Is x-y ≠ 0 transitive?"
I think it is transitive since
x - y ≠ 0
y -...
1
vote
0answers
17 views
Composition of jets
I was reading that for a function $f:\mathbb{R}^m\rightarrow \mathbb{R}^n$ such that $f(0)=0$ and a function $g:\mathbb{R}^n\rightarrow \mathbb{R}^d$, $J(g\circ f)=J(g)\circ J(f)$ where $J(f)$ denotes ...
0
votes
0answers
13 views
Chain rule of higher derivative with multivariable functions
I wanted to know if there is a general formula in order to calculate $\frac{\partial^{k}f\circ g}{\partial x_{i_k}\dots \partial x_{i_k}}$ using $\frac{\partial^{d}f}{\partial x_{j_d}\dots \partial x_{...
1
vote
1answer
19 views
Composition of an inverse function with another function
Suppose I have two functions $f(x)$ and $g(y)$. Then what is $f^{-1}$ composed with $g$, i.e.:
$f^{-1}\circ g$ ?
To me it looks like the value of $x$ when $f(x) = g(y)$, but I am not entirely sure. ...
2
votes
1answer
31 views
Function composition on a commutative diagram: basic question
I'm trying to work through Categories for the Working Mathematician and immediately ran into a point of confusion.
A composition of functions is commutative iff $f\circ g = g\circ f$. But if in this ...
1
vote
1answer
30 views
An extra condition for composition of relations which are not necessarily functions
Given two binary relations $\mathcal R$ and $\mathcal S$ over $A \times B$ and $B \times C$,then their composition relation is denoted by $\mathcal S \circ \mathcal R$ and is defined as:
$$\mathcal S \...
-1
votes
1answer
20 views
Finding the Integral of A Composition of Logarithm, exp, and Trigometric Functions
I have the function $$f(x)=\ln \left(e^{5+6\left(\sum_{k=0}^{100} \frac{\sin \left(k\pi x\right)}k \right)}+e \right)$$
One, is it possible to integrate this without "special functions" (...
2
votes
2answers
60 views
Finding a variable $n$ that satisfies the functional composition in which $f(f(f(f(n))))=3.$
Let $f(x) = x^2-2x$. How many distinct real numbers $c$ satisfy $f(f(f(f(n)))) = 3$?
I have not found any good way to do this problem. I have just resorted to start with brute force:
We start with $$...
2
votes
2answers
56 views
A binary relation $\mathcal R$ over a set $A$ is transitive if and only if $\mathcal R$ is equal to its transitive closure $\mathcal R^{+}$.
Given a binary relation $\mathcal R$ over a set $A$,then the $\mathsf {Transitive \;Closure}$ of $\mathcal R$ over $A$ is the smallest transitive relation on $A$ containing $\mathcal R$, it's indeed ...
1
vote
2answers
44 views
Conjecture about the transitive closure of a relation $\mathcal R $ over a finite set $A$
Given a binary relation $\mathcal R$ over a set $A$,then the transitive closure of $\mathcal R$ over $A$ is the smallest transitive relation on $A$ containing $\mathcal R$, it's indeed the ...
1
vote
0answers
49 views
Partial derivatives of a multivariable composite function (chain rule)
Say I have this function:
$$z=f(ax, ay)$$
I know how to take this partial derivative by defining $u=ax$ and $v=ay$:
$$\frac{\partial z}{\partial x}=\frac{\partial f(ax, ay)}{\partial x}=\frac{\partial ...
0
votes
1answer
41 views
Decomposition of a function and chain rule.
This question is about the basic chain rule (and I think of it when I read about calculation of variation in defining distance in manifold using usual Riemannian metrics) and is related to the another ...
5
votes
1answer
58 views
Can we really compose random variables and probability density functions?
A renowned professor of statistics (whose name I will not reveal here) told me that the notation $p(x)$ makes perfect sense when $p$ is a pdf and $x$ is a RANDOM variable (i.e. a function). I was a ...
0
votes
1answer
26 views
Composition Functions (Advanced Functions)
Question: a) Given the functions $f(x) = x + 2$ and $g(x) = 3^x$, determine an equation for (f ∘ g)(x) and (g ∘ f)(x).
b) Determine all values for $x$ for which $f(g(x)) = g(f(x))$.
*For part a), I ...
0
votes
2answers
34 views
Composing trigonometric functions
Let $f(x)=\sin(x)$. If $g$ and $h$ are functions on $\mathbb{R}$ such that $g(f(x))= h(f(x))$,
can we conclude $g=h$ ?
Can we actually compare $g$ and $h$?
I am confused. Please, help me.
0
votes
1answer
48 views
Questions about relation composition and equivalence relation
Let $R \subseteq S\times S$ be any binary relation on a set S. Consider the sequence $R^{0}, R^{1}, R^{2}...$ defined as follows:
$$R^{0} := I = \left \{ (x,x):x\in S \right \}\\
R^{n+1} :=R^{n}\cup \...
3
votes
1answer
29 views
Continuity with composition functions
Is there a continuty in $g(f(x))$ at $x = 4$?
$$f(x)=\begin{cases} -1 &, x=4 \\ 1 &, x \ne 4\end{cases}$$
$$g(x)=\begin{cases} -6 &, x=4 \\ 4x-10 & x \ne 4\end{cases}$$
My answer for ...
0
votes
1answer
85 views
Is $f^2 \circ f=f \circ f^2$ true?
Suppose $f:\mathbb{R} \rightarrow \mathbb{R}$ is a function from the set of real numbers to the same set with $f(x)=x+1$.
We write $f^{2}$ to represent $f \circ f and f^{n+1}=f^n \circ f$.
Is it true ...
0
votes
1answer
20 views
Need the result of composing an infinite number of smooth functions be smooth?
$f$ is a smooth function from a manifold to itself. So is $f\circ f$, and $f\circ f\circ f$ and so on...
If this sequence is extended forever, and supposing that it converges to some function, need ...
2
votes
1answer
79 views
Computing $f \circ g$ and $g \circ f$ for functions by cases
I want to confirm my solution for the following problem.
Problem: Let $f,g: \mathbb{R} \rightarrow \mathbb{R}$ be given by
$$
f(x) = \begin{cases} 1-2x, & if & x\geq 0 \\
\left|x\right|, &...
3
votes
1answer
52 views
For an infinite sequence of functions $\Bbb{R}\to\Bbb{R}$, each function is a composition of a certain finite set of functions $\Bbb{R}\to\Bbb{R}$.
Given an infinite sequence of functions $\{g_1, g_2, \ldots, g_n, \ldots\}$ where $ g_n : \Bbb R \to \Bbb R$ prove there's a finite set of functions $ \{ f_1, f_2, \ldots, f_M \} $ such that any $ g_n ...
0
votes
0answers
43 views
Injectivity proof of multiple function composition
Given two functions $h(x)$ and $f(x)$, with $x \in X \subset \mathbb{R}^6$, with the first $3$ elements of $X$ being a unit vector.
I want to prove that a map $F$ given as
$F = \begin{bmatrix} h(x) \\ ...
0
votes
0answers
24 views
Doubt based on composition of functions
I have a question regarding composition of functions, which goes as follows:
Let two functions are defined as
$$g(x)=\begin{cases}
x^2, &-1\leq x<2 \\
x+2, & 2\leq x\leq 3\\
\end{cases}$$
...
0
votes
1answer
17 views
Graph of this function
Let f is a real valued function defined from R to R such that f(x)+f(-x)=5
Is this function even , can we plot this function ‘f’ on graph?
And what information do we get from this functional equation?...
0
votes
1answer
34 views
if $f(x)=x^2+x+2$ & $g(x)=x^2-x+2$, How to prove that there is no fuction $ h:\mathbb R \to \mathbb R $ exist such as $h(f)+h(g) = g(f)$?
I tried to substitute $f(x)$ & $g(x)$ in their places but didn't find a relation;
The function beginning bijective or surjective etc have nothing to do with our case I believe
$g(f(x))=x^4+2x^3+...
0
votes
2answers
41 views
Find linear transformations $U,T:\textbf{F}^{2}\to\textbf{F}^{2}$ such that $UT = T_{0}$ (the zero transformation), but $TU\neq T_{0}$.
Find linear transformations $U,T:\textbf{F}^{2}\to\textbf{F}^{2}$ such that $UT = T_{0}$ (the zero transformation), but $TU\neq T_{0}$.
My solution
Let us consider $T(x,y) = (x,0)$ and $U(x,y) = (y,...
0
votes
1answer
19 views
Demonstrate the truthfulness of the statement
I have the following statement:
Determine if is true that if $g: \mathbb{R} \to \mathbb{R}, f: \mathbb{R} \to \mathbb{R}$ and $ (g\circ f)(x) = x$ therefore $g = f^{-1}$
My attempt was:
$i)$ ...
2
votes
1answer
23 views
Functional equation involving composition and exponents
Do there exist functions $f,g : R → R$ such that $f (g(x)) = x^2$ and $g( f (x)) = x^3 \text{ , }\forall x ∈ R$.
Simply applying $g$ on both sides of equation $1$ and $f$ on equation $2$ ...
