Mathematics Stack Exchange
Mathematics Stack Exchange

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 $S\circ R=\{(a,c); (\exists b\in B) (a,b)\in R \land (b,c)\in S\}$.

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On the functional square root of $x^2+1$

There are some math quizzes like: find a function $\phi:\mathbb{R}\rightarrow\mathbb{R}$ such that $\phi(\phi(x)) = f(x) \equiv x^2 + 1.$ If such $\phi$ exists (it does in this example), $\phi$ can be viewed as a "square root" of $f$ in the sense…
user1551
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Find three non-constant, pairwise unequal functions $f,g,h:\mathbb R\to \mathbb R$...

I've been stumped by this problem: Find three non-constant, pairwise unequal functions $f,g,h:\mathbb R\to \mathbb R$ such that $$f\circ g=h$$ $$g\circ h=f$$ $$h\circ f=g$$ or prove that no three such functions exist. I highly suspect, by…
Franklin Pezzuti Dyer
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What numbers can be created by $1-x^2$ and $\frac{x}{2}$?

Suppose I have two functions $$f(x)=1-x^2$$ $$g(x)=\frac{x}{2}$$ and the number $1$. If I am allowed to compose these functions as many times as I like and in any order, what numbers can I get to if I must take $1$ as the input? For example, I can…
Franklin Pezzuti Dyer
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Which functions satisfy $f^n(x) = f(x)^n$ for some $n \ge 2$?

Let $n$ be an integer greater than $1$. The notation $f^n$ is notoriously ambiguous: it means either the $n$-th iterate of $f$ or its $n$-th power. I was wondering when the two interpretations are in fact the same. In other words, if we write…
Luca Bressan
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Given $g(x)$ and $f(g(x))$, solve for $f(x)$.

I've hit a wall on the above question and was unable to find any online examples that also contain trig in $f(g(x))$. I'm sure I am missing something blatantly obvious but I can't quite get it. $$ g(x)=3x+4 , \quad f(g(x)) =…
Dwayne H
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Characterising functions $f$ that can be written as $f = g \circ g$?

I'd like to characterise the functions that ‘have square roots’ in the function composition sense. That is, can a given function $f$ be written as $f = g \circ g$ (where $\circ$ is function composition)? For instance, the function $f(x) = x+10$ has…
bryn
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Show that Function Compositions Are Associative

My intent is to show that a composition of bijections is also a bijection by showing the existence of an inverse. But my approach requires the associativity of function composition. Let $f: X \rightarrow Y, g: Y \rightarrow Z, h: Z \rightarrow…
Andy Tam
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composition of two uniformly continuous functions.

Let $f : \mathbb{R} \rightarrow \mathbb{R}$ and $g : \mathbb{R} \rightarrow \mathbb{R}$ are two uniform continuous functions. Which of the following options are correct and why? $f(g(x))$ is uniformly continuous. $f(g(x))$ is continuous but not…
Supriyo
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Are there two functions $f, g$ such that $f(g(x)) = x^3$ and $g(f(x)) = x^5$?

Question. Are there two functions $f, g: \mathbb{R}\rightarrow\mathbb{R}$ that satisfy $f(g(x)) = x^3 \enspace\forall x\in\mathbb{R}$ and $g(f(x)) = x^5\enspace\forall x\in\mathbb{R}$? This is an extension to this question, where I proved that…
Maximilian Janisch
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half iterate of $x^2+c$

I'm looking for literature on fractional iterates of $x^2+c$, where c>0. For c=0, generating the half iterate is trivial. $$h(h(x))=x^2$$ $$h(x)=x^{\sqrt{2}}$$ The question is, for $c>0,$ and $x>1$, when is the half iterate of $x^2+c$ smaller than…
Sheldon L
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Real Analysis: Continuity of a Composition Function

Suppose $f$ and $g$ are functions such that $g$ is continuous at $a$, and $f$ is continuous at $g(a)$. Show the composition $f(g(x))$ is continuous at $a$. My idea: Can I go straight from definition and take $\delta=\min\{\delta_1,\delta_2\}$, where…
Mr.Fry
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Why any countable subset of $\mathbb{R}→\mathbb{R}$ is generated by a finite set under composition?

Given a sequence of functions $\{g_k\}$, where $g_k: \Bbb R\to\Bbb R$ for all $n\in \mathbb N$. Prove that there exists a finite set of functions $$f_1,f_2,\ldots,f_n$$ such that any function $g_k$ can be expressed as a composition $$f_{k_1}\circ…
Nikolay Chistyakov
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Crazy Iterated Square Roots

I was messing around with infinite square root nesting problems like $$w_1=\sqrt{1+\sqrt{1+\sqrt{...}}}$$ which is an easy example. I decided to try one where the terms inside of the square roots form a geometric…
Franklin Pezzuti Dyer
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Number of real zeroes of iterated polynomial: $x^3-2x+1$

If $P(x)=x^3-2x+1$, define $z_n$ as the number of real roots of the polynomial $P^{\circ n}(x)$, where the superscript denotes $n$-fold composition. Can we find a general formula for $z_n$, or perhaps a recurrence of some sort? The sequence begins…
Franklin Pezzuti Dyer
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$f(f(x))=f(x)$ question

I am wondering what is the class of functions $f: \mathbb{R}\rightarrow\mathbb{R}$ such that $f(f(x))=f(x)$? I think it should be: Constant Value functions the identity function absolute value function $|x|$ But I don't know if this is right or…
User112358
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