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Questions tagged [function-and-relation-composition]

For questions about the composition of functions and relations.

For questions about the composition of functions and relations: If $f\colon A\to B$ and $g\colon B\to C$ are functions, their composition $g\circ f\colon A\to C$ is given by $x\mapsto g(f(x))$. Similarly for $R\subseteq A\times B$ and $S\subseteq B\times C$ the composite relation is defined $S\circ R=\{(a,c); (\exists b\in B) (a,b)\in R \land (b,c)\in S\}$.

1202 questions
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If the absolute value of a function is continuous, is the function continuous?

If $|f(x)|$ is continuous at $a$, is $f(x)$ continuous at $a$? I tried doing it using composite functions. If $g(x)= |x|$, then $g\circ f(x)= |f(x)|$. Since $g(x)$ and $g\circ f(x)$ are continuous, $f(x)$ is continuous. I don't know if this is…
Gauri Sharma
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How do I find a function g that yields a specific answer when composed with f?

Let $f(x)=\frac x{x-2}$ . Find a function $y=g(x)$ so that $(f\circ g)(x)=4x$. The solution is $g(x)=\frac{8x}{4x-1}$, but I don't know how to arrive at this answer. Please help me to find the steps to solve this problem.
error
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Can any function be written as a composition of other functions?

Can any function be written as a composition of other functions? For example, can a polynomial $h(x)$ be written as $k(g(x))$?
Farman H
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If $f:\mathbb{R} \to \mathbb{Z}$, why can't we have that $g \circ f$?

In the lecture notes for a course I'm taking, it is stated that: Unlike multiplication however, we can't reverse order. First of all, in general it doesn't even make sense to reverse composition. For example, if we have $f:\mathbb{R} \to…
Chelsea Kelly-Reif
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If $f \circ g = g \circ f$ but $f$ and $g$ are NOT inverses or equal, can anything general be said about $f$ and $g$?

If $f$ and $g$ are both linear, it's not difficult to find examples, e.g. $f(x)=7x-3$ and $g(x)=11x-5$; in general, for $f(x)=ax+b$ and $g(x)=cx+d$, the necessary and sufficient condition is easily seen to be $\frac{a-1}{c-1}=\frac{b}{d}$. For…
A.J.
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Show that $n^{4}-20n^{2}+4$ is a composite whenever $n$ is an integer

Show that $n^{4}-20n^{2}+4$ is a composite whenever $n$ is an integer. I'm having trouble putting together a proof for this problem.
user61646
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Find inverse composition

I've been stuck for an hour trying to solve this composition function: If $f(\sqrt[7] x)=g(3x-5)$, then $[(g^{-1}\circ f)^{-1} \circ g^{-1}](x)=\ldots$ My friend gave me a hint that I should find $f^{-1}$ first, but I don't really get it. Can…
Znik Dzulqarnain
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Find all $g(x)$ such that $f(g(x))=g(f(x))$

For a given function $f(x)$ I want to find all functions $g(x)$ such that $f(g(x))=g(f(x))$. Two solutions are always $g(x)=x$ and $g(x)=f(x)$, but are there any more? Initially I wondered this for $f(x)=e^x$, but then also wondered about the…
Kwin van der Veen
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Is it possible to find $g^k(x)$?

Given $g(x)=\frac{x}{2}+\frac{1}{x}$, is it possible to find an expression for $g^k(x)=(g\underbrace{\circ \cdots \circ}_k g)(x)$, where $k$ is some positive whole number? For example, given $h(x)=\frac{x}{2}$, the expression for $h^k(x)$ is…
user299248
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Can the composition of two non-invertible functions be invertible?

(Context: I came across this exercise in the textbook "Coding the Matrix" when reading it to supplement my studies in the Coursera class "Coding the Matrix".) After proving that the composition of invertible functions is itself invertible (by…
Aneesh Mulye
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Function composition proof

One of the questions assigned to me for homework was: Is it true that $ f\circ (g \circ h ) =f \circ g + f \circ h$? I am in the understanding that $f \circ g$ means $f(g(x))$, so $ f\circ (g \circ h )$ is $f(g(h(x)))$. Well let us take $f(x)=c$…
Jori
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Is this a valid example for $f \circ g = id_M$, but $g \circ f \ne id_N$

The exercise is to give an example for two sets $M$ and $N$, and functions $f$ and $g$, for which $f \circ g = id_M$, but $g \circ f \ne id_N$. My idea is a bit based on my computer programming background, where (x/2)*2 is 0 for integers. Here it…
phresnel
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Discontinuous commuting function

Can two commuting (composition of the functions satisfies commutativity) with $f\ne g$ and both $f$,$g$ increasing functions on $[0,1]$ both be discontinuous on the set of rationals? Context: I had recently asked (and subsequently removed) a…
Rex Nolan
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On existence of composition of functions

Let $f: A \rightarrow B$ and $g: B \rightarrow C$ be two functions. Then $g \circ f$ is clearly defined $\forall a \in A$ but what about $f \circ g$, do we take it as undefined, given that A and C are disjoint? What if A and C are not…
Aditya
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Calculating composition - What is this function?

I am working on an assignment for my advanced math class. I have the following question: The question goes on but I am having trouble understanding what this part of the question is asking. I understand that I am determining G composed of…
Doug Ramsey
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