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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why isn't this function surjective?

I have two functions, $f:\mathbb{Q}\to\mathbb R_+$ given by $f(x) = \frac{1}{5} \sin{\left (\pi x \right )} + 4 $, and $g:\mathbb{Z_+}\to\mathbb{Q}$ given by $g(x) = - \frac{7 x}{2}$. I needed to find the composite function $h(x)=f(g(x))$ and I…
Simon
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Finding $g(x)$ , given $h(x)$ and $f(x)$

find $g(x)$ $f(g(x)) = h(x)$ $f(x) = x^{2}-6x+2$ $h(x) = 9x^{2}-24x+9$ My work: $g(x)^{2}-6(g(x))+2 = h(x)$ $g(x)^{2}-6(g(x))-(9x^{2}-24x+7)$ And then what's next step?please provide explaination as well Thank You!! 3+- sqr(9-(-9x^2-24x+7)) 3+-…
Secret
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Injective Maps and Preimages

Let $X$, $Y$ be sets and let $f:X \to Y$ be a map. For any subset $C \subseteq Y$ of the codomain $Y$, we consider its preimage $f^{−1}(C) \subseteq X$ in the domain $X$: $f^{−1}(C) := \{x \in X : f(x) \in C\}$. (Note: the map $f$ here is not…
user271716
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Taking the compositions of two constant functions

The questions asks to prove that the composition of g with f is not equal to f with g. However, I don't know whether you can even take the composition of constant functions or how. so if f(x)=2 and g(x)=4. How do you take the composition?
Undergradstudent
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$n$ composition of functions when $n \to \infty$

Let $x \in \mathbb{R}$ and $n \in \mathbb{N}$. Let $f(x)$ be continous over the whole domain of $a
Erivelton Geraldo Nepomuceno
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Solve the composition $f \circ g=?$ and $g\circ f=?$

a) $f(x) = \sqrt[3]{x}\ $ and $g(x) = x^3$ find $f\circ g=?$ and $g\circ f=?$ I have $f\circ g = f(g(x)) = f(x^3) = \sqrt[3]{x^3} = x$ So basically, first we replace $g(x)$ with its value, then we replace $x$ with the value of $f(x)$ inside…
user95523
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Name of a family of bijections among a family of sets that is closed under composition

I am considering a structure composed of a set of sets, $\{S_i\}$, and a set of bijections between some pairs of those sets, $\{f_{ij}\}$, where each $f_{ij}$ that exists is a bijection from $S_i$ into $S_j$. One can then attempt to extend the set…
Dale
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Generalizing a proof about a preserved property under composition

There is a property of binary operations (functions from $\mathbb{S}^2$ to $\mathbb{S}$ for an arbirtary set $\mathbb{S}$) that I'm trying to figure out whether or not it is preserved. The cleanest way I know to describe it is: A function…
Samuel Owen
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Composition of mappings, composition table

I have a several questions on the following problem. Let X = {1,2}. There are 4 mappings from set X to itself, denoted End(X) and all the compositions betwen them are defined. If we denote f $\in$ End(X) as a 2-letter word (f(1), f(2)) (note:…
John Doe
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Suppose $f$ and $g$ are both functions from $A$ to $A$. If $f \circ f = g \circ g$, does it follow that $f = g$?

We know that $(f(f(a)) = g(g(a))$ for some $a \in A$. I've tried and tried, both with a finite and infinite $A$, but I can't seem to find a counterexample wherein $f$ and $g$ are not the same. Apologies for the simple question, but I'd appreciate a…
Eragon20
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Why a composite functions is well defined with this conditions?

I have read on many sites(per example this MSE' answer) that a $gof$ composite function is "well defined" or "is possible" when: $f: A \to B$, $g: B \to C$, $gof:A\to C$ What this mean? is definition or what? Why the domain of $gof$ is $A$? it is…
ESCM
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Prohibited values when plugging one formula into another?

Say f(x)=$\frac{1}{x}$ and g(x)=$x-5$ and we where to plug g into f as a composite function, would x = 0 still remain a prohibit value of f or not? Thanks!
Pregunto
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Function composition - solving for functions

I have a simple, continuous, real-valued function $\sigma$, whose functional form I know, and I know that two other invertible functions, $f$ and $g$, satisfy the following relationships: $f\circ\sigma\circ\sigma\circ…
AnotherShruggingPhysicist
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Unable to understand how to solve piecewise composition of two functions

$$f(x)= \begin{cases} 2x & ; x < 0 \\ \sqrt{x} & ; 0 \leq x \leq 1 \\ (x-1)^2+1 & ; x > 1 \end{cases} $$ $$g(x)= \begin{cases} x^2 & ; x \leq 1 \\ 1 &; x > 1 \end{cases} $$ Hello, firstly I would like to thank everyone who is reading this post.…
tone tinkovič
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(f o g) o f composition function

If $f$ represents the function $f(x)$ and $g$ represents the functions $g(x)$, is $(f \circ g)\circ f$ essentially $f(g(f(x)))$? I know that: $(f \circ g) = f(g(x))$ however I'm not sure if the brackets in my equations make a difference to this…
Jessica Tiberio
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