# Tagged Questions

For questions about the composition of functions: 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 $x\mapsto g(f(x))$.

24 views

### How to solve diff. eq. involving function taking expression including itself as variable

$$f(x)=f \left( x \pm \frac{l}{\sqrt{1+\dot{f}^2}} \right) \mp \frac{l}{\sqrt{1+\dot{f}^2}}\dot{f}^2,$$ where $l$ is a constant. How is such a beast even approached? If anyone got intuition for ...
50 views

### Suppose $f(x)$ and $g(x)$ are $1$−$1$ functions on their respective domains. Show that $f(g(x))$ is a $1$−$1$ function.

I have an idea of where to go with this proof, but would like a second opinion as to wether I have actually made a logical argument. $f(g(x)) \neq f(g(y))$ where $x, y \in \mathbb{R}$ and $x \neq y$. ...
57 views

25 views

### Derivatives of Implicit Functions (Abstract Case)

I have never been good at differentiation of implicit functions in cases when in a function is given, much less in abstract cases with composite functions. Hopefully someone can help me get started on ...
32 views

### Why is this proof for an arbitrary function constrained to a constant one?

Sorry if this seems trivial, I'm having some difficulty understanding a proof. I'm doing exercise 5.1.14 of Velleman's How to Prove It and a solution posted in this question, including the comments, ...
45 views

### Compose $(1243)$ and $(5)$

Checking my work. In either direction: $(1243)[1] = 2$ and $(5)[2] = 2$, so far we have $(1, 2,\ldots$ $(1243)[2] = 4$ and $(5)[4] = 4$, so far we have $(1, 2, 4,\ldots$ $(1243)[4] = 3$ and ...
53 views

### Composition factoring into nonnegative integer polynomials

Consider the integer polynomials with nonnegative coefficients, such as: $$1 + 2x +2x^2$$ $$3 + 3x^4 + 11x^{10}$$ $$13 + 7x + x^2$$ I asm interested in knowing "what is a composition ...
59 views

### General closed form solution to $f'(x) = P(f(x))/P(x)$

Does there exist a general closed form solution (in terms of elementary or special functions) to the differential equation: $$\frac{df(x)}{dx} = \frac{P(f(x))}{P(x)}$$ when $P(x)$ is a polynomial ...
38 views

### Composition of analytic functions is analytic

I want to find a proof that shows the composition of two analytic functions is analytic. I know I should prove this using Cauchy-Riemann equations, but I wasn't able to use them in the proper way in ...
16 views

### Matrix representation of function concatenation using other basis than polynomials.

I have now familiarized myself with the Carleman-matrices which represent function composition of polynomials (actually taylor series terms) and built some of my own. I noticed that for any finite ...
26 views

### Function composition as representable by matrices?

I know from linear algebra that for different sets of functions differentiation can be expressed using matrix multiplication on a vector representation of the function. For instance polynomials and ...
58 views

### The limit of a composed function

Let $g_n: \mathbb{N} \rightarrow \mathbb{R}$ and $f_n(x): \mathbb{N\times R} \rightarrow \mathbb{R}$. If $g_n \rightarrow g$ and $f_n(g) \rightarrow f(g)$, can we deduce $f_n(g_n) \rightarrow f(g)$? ...
65 views

### Composite Function Equality

Let f, g be functions$: \mathbb{R} \mapsto \mathbb{R}$ with $f(x)=x^2+ax+b$ with $a,b \in \mathbb{R}$, such that $(f\circ g)=(g\circ f)$. If the equation $g(x)=x$ has precisely one solution for all ...
53 views

### Composition operators on fractional-order Sobolev spaces

Preliminaries: We know that the fractional-order Sobolev spaces $\mathrm{H}^s(\mathbb{R})$ and $\mathrm{H}^s(\mathbb{T})$ are closed under multiplication provided $s > 1/2$. This is proved for ...
45 views

16 views

### How to determine a function from a sequence of consecutively composed functions?

Let $f(x) = x+1$ and $g(x) = 2x$ Prove $f^2g = gf$ and determine $f^igf^jgf^k(x)$ explicitly as a function of x and in terms of i,j,k. I got through the proof but I don't understand what the ...
35 views

### If a composition of two maps is smooth, as well as one of the maps, then so is the other.

Let $M$, $N$, and $K$ be smooth manifolds, and consider the maps $g:M\to N$, and $f:N\to L$. Assume that the composition $f\circ g$ is smooth. If any of $f$ and $g$ is smooth can we conclude that the ...
32 views

### Permutations, compositions and associativity properties

Let n be a postive integer, and let σ : {1, . . . , n} → {1, . . . , n} be a one-to-one and onto map. Then σ is called a permutation on n elements. The set of all permutations on n elements is denoted ...
186 views

### The composition of a nowhere-differentiable function with a differentiable function.

This is actually Problem $17$ from Chapter $10$ of the Fourth Edition of Michael Spivak’s Calculus. The statement is quite simple, but I have not had any success in finding an example. Here is the ...
92 views

### If $f$ is injective and $g$ is injective, then $f \circ g$ is surjective. [duplicate]

I can prove that if $f$ and $g$ are both injective, then $f \circ g$ is injective, but I don’t know how to prove that $f \circ g$ is surjective.
49 views

### Show that $g$ is one-one if and only if $g$ is onto.

Original problem A function $g$ from a set $X$ to itself satisfies $g^m=g^n$ for positive $m$ and $n$ with $m>n$. Here $g^n$ stands for $g\circ g\circ \dots g$(n times). Show that $g$ is one-one ...
19 views

### Taylor Expansion of Composition of Functions

I have in my lecture notes that $$f(g(x+h))-f(g(x))\approx f'(g(x))\left(g(x+h)-g(x)\right)+f''(g(x))(g(x+h)-g(x))^2$$ He explained can found via taylor expansion, but I try to expand it and am not ...
45 views

### Linear transformation of eigenspace is subset of eigenspace

Let $V$ be a vector space over a field $\mathbb{F}$ and let $L$, $M$ be two linear transformations from $V$ to itself. a. Show that the subset $W= {x ∈ V : L(x) = M(x)}$ is a subspace of $V$ b. ...
42 views

### Composition of 2 monotonic functions

Let $f$ be a monotonic function $f:[a,b] \rightarrow\mathbb{R}$ and $g$ be a monotonic function $g:[c,d]\rightarrow[a,b]$. Show that $f\circ g$ is monotonic
20 views

### Given two piece-wise functions, prove their composition is periodic.

I know how to graph h(f(x)), however, I am having trouble proving it is periodic. i. We must prove that h(f(x + k)) = h(f(x)) for all x ∈ ℝ and for some k ≠0 ∈ ℝ in order to prove that h(f(x)) is ...
30 views

### $f \circ f^{-1} = i_B$ proof using the fact that $f^{-1} \circ f = i_A$

Suppose f is function from A to B, and suppose that $f^{-1}$ is a function from B to A. Assume $f^{-1} \circ f = i_A$. Then show therefore that $f \circ f^{-1} = i_B$. I tried applying left ...
78 views

### We all know about compositions of functions, but what about decomposition. Is there a way with math, not just heuristics?

The composition operator is a well know and quite often used method in integration and differentiation, think u-substitution. However, given a composition like $$f(f(f(...f(x)...)))$$ Where there are ...
31 views

### functions compositions, three sets, counting compositions

Given three sets $P, Q, R$ such that $|P|=p, |Q|=q, |R|=r,$ and $p,q,r > 1$ let $f(x): P\rightarrow Q$, and $g(x):Q\rightarrow R$ be two functions. Find the number of functions which can be ...
### How to find $s(\exp(d(x)))$ ~ $x + 2$?
Let $x$ be a positive real. I want to find a pair of analytic functions $s(x),d(x)$ such that $s(d(x)) = x$ and $s(\exp(d(x)))$ ~ $x + 2$ More presicely I Also want :  \lim_{x \to \infty} ...