# Questions tagged [function-and-relation-composition]

For questions about the composition of functions and relations.

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### How to understand such a function $g(I+X)$?

Recently I've been reading Lie Groups written by Daniel Bump. Lemma 7.1. Let $f$ be a smooth map from a neighborhood of the origin in $\mathbb{R}^n$ into a finite-dimensional vector space. We may ...
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### $f$ and $g$ are functions from $\mathbb{R} \to \mathbb{R}$. If $f(x) = 2x+6$ and $g(x)= x^3$, what is $g\circ f$?

I was able to get the first part of this question which was $f\circ g = 2x^3+6$. I was unable however to answer the question which I posted.
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### Confusion regarding the definition of composition of functions

This question may seem kinda silly but in constructing a well organized proof about the associativity of function compositions I need to clear my confusion. Here's the definition of composition of ...
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### This function has a fixed point, $F^4(x)=x$. Why?

Consider two points on the unit sphere, $C_1$ and $C_2$. Let these points be close enough such that circles of radius $r$ drawn around each point intersect at two points, $N_1$ and $N_2$. The vectors ...
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### systems of equations involving composition of functions

If we are given a function $g(x) = x - 1/x$ And another one given in terms of composition $f(g(x)) = x^3 - 1/x^3$ By which general method does one find $f(x)$ ? Can it be found for arbitrary $g(x)$ ...
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### Why codomain is more than the range in an Inverse function

While solving inverse function problems, I got confused in a part, like for any Inverse function to be defined, it must be one-one and onto, then in many questions why the codomain is given more than ...
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### From the discontinuity of $f(x)$ and $g(x)$, can we directly tell about the discontinuity of $f(g(x))$?

From the discontinuity of $f(x)$ and $g(x)$, can we directly tell about the discontinuity of $f(g(x))$? I thought $f(g(x))$ would be discontinuous where $g(x)$ is discontinuous and where $g(x)=c$, ...
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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 ...
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### Cardinality of a collection of functions whose composition commutes

Consider a collection of functions $\mathcal F$ where (i) each individual element is a strictly increasing function from [0,1] to [0,1]; (ii) for any $f,g$ $f \le g$ or $f \ge g$ and $f\ne g$; and (...
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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 ...
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### Is a norm of a vector of distances itself a metric on vectors?

Suppose I have $\vec x = (x_1, \ldots, x_n)$ and $\vec y = (y_1, \ldots, y_n)$ and a sequence of metrics $d_i$ for $i \in \{ 1, \ldots, n\}$ that are used for the $i$th component. Consider a vector of ...
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### Show that if $g\circ f$ is injective, then $f$ must be injective. Is it true that $g$ must also be injective? [duplicate]

I am now self-studying Terence Tao's Analysis 1. I am trying to solve all of the exercises. The question I have a problem with is Let $f: X\rightarrow Y$ and $g:Y\rightarrow Z$ be functions. Show ...
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### How to intuitively think about composition of relations?

Examples: $A \cap B$ elements that are in both $A$ and $B$ $A \cup B$ elements that are in either $A$ or $B$ $R \subset A\times B$ coordinates where the first coordinate is an element of $A$, and ...
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### Is $f(g(x))$ discontinuous?

Question: Let $f(x) = \frac{1}{15x^2+8x+1}$ and $g(x)= \frac{1}{(x-1)(x-2)}$, then the number of points of discontinuity of $f(g(x))$ is? The answer key claimed that the answer is $1$, but I don't ...
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### Low rank function decomposition — necessary and sufficient conditions

Given a function ${\bf f} (x,y,z) = (f_1, \dots, f_n) : \mathbb{R}^3 \to \mathbb{R}^n$, suppose I can write ${\bf f} (x,y,z) = ({\bf g} \circ p)(x,y,z)$ where $p(x,y,z) : \mathbb{R}^3 \to \mathbb{R}$ ...
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### If $h:[0,1]\to [0,1]$ is continuous and surjective, then does there exist a continuous function $f$ such that $ff=h?$ If so, is $f$ unique?

Let $h:[0,1]\to [0,1]$ be a surjective continuous real function. There surely exist many functions $f:[0,1]\to [0,1]$ such that $f(f(x)) = h(x)\$ on $x\in [0,1].$ Does there exist a continuous ...
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### Vector-valued function composition

I was going through the blog article at Matrix Calculus. Under the section Vector chain rule they did some function composition thing to demonstrate the chain rule. The example that they took is as ...
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