Elementary questions about functions, notation, properties, and operations such as function composition.

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26 views

If $f(x)=\sin(\log(\frac{\sqrt{4-x^2}}{1-x}))$ then the range of $f(x)$ is? [duplicate]

If $f(x)=\sin(\log(\frac{\sqrt{4-x^2}}{1-x}))$ then the range of $f(x)$ is? I found the domain of the function is $-2<x<1$.But I'm having difficulty in finding the range.
2
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1answer
70 views

How to formalize a variable-binding operator, such like $\frac{d}{dx}$?

How to formalize a variable-binding operator, such like $\frac{d}{dx}f(x)$? For instance, I think we should treat $\frac{d}{dx}$ as a higher-order function of $x$, returning a function that takes it ...
1
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1answer
67 views

Have I done this correctly?

I need to show that $\sum_{n=1}^{\infty}\frac{(-1)^{n+1}}{n+x^2}$ converges uniformly but not absolutely on $\mathbb{R}$. First, I showed that the absolute value of the partial sums diverges for ...
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1answer
10 views

Factors/divisibility of monotonically-increasing integer polynomial

For positive integers $n$ and $x$, let $f_n(x)$ be a polynomial in $x$ of degree $n-1$, such that $f_n(x)$ is monotonically increasing for increasing $x \ge 1$. Now assume that there exist positive ...
4
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2answers
404 views

Forcing Bijectivity

I'm working out of the Nakahara text in mathematical physics, and I'm presented with a map $f: \mathbb{R} \rightarrow \mathbb{R}$ defined by $ f:x \mapsto \sin(x) $, and told that it is neither ...
0
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1answer
19 views

If the dimension of the domain of a linear transformation is smaller than that of the codomain, the transformation is not surjective always?

Given two vector spaces $V$ and $W$, defined over the same field $K$, and a linear transformation between them $T:V\longrightarrow W$, such that $\text{dim}(V) = n$, $\text{dim}(W) = m$ and $n \lt m$, ...
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2answers
24 views

Suppose $f(x)=x(x-2)$ and $h(x)=e^{-2x}$. If $f(a)+h( \ln a ) =0$, show that $a^3-a^2-a-1=0$

Suppose $f(x)=x(x-2)$ and $h(x)=e^{-2x}$. If $f(a)+h(\ln a ) =0$, show that $$a^3-a^2-a-1=0$$ My attempt: $a(a-2) + \dfrac{1}{a^2}=0 \Rightarrow a^4-2a^3+1=0 \Rightarrow a^3 = \dfrac{1}{2-a}$ ...
0
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1answer
26 views

If a curve has $2$ stationary points, then $a<0$ or $a>3$

The curve $C$ has equation $y=ax+a+\dfrac{a-3}{x-1}$, where $a$ is a non-zero constant. Prove that if $C$ has two stationary points, then $a<0$ or $a>3$. My attempt: From the equation of the ...
1
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0answers
36 views

How to denote a variable is an argument to a function.

How would one write "x is an argument to the function f" in set notation. For instance here is a piece of logic I'm trying to write as set notation: For all x where x is an argument to the function ...
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1answer
17 views

Proving injectivity of a piecewise defined function; slight conceptual issue

I am having a slight issue wrapping my head around a specific concept in the following problem: Let $A,B,C,D$ be sets such that $A$ and $B$ are disjoint and $C$ and $D$ are disjoint. Let $f:A\to C$ ...
3
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1answer
99 views

Find monotonic functions going from $0$ to $+\infty$ for $x \in (-\infty,+\infty)$ (similar to $e^x$)

How can we find functions on $\mathbb{R}$ with exponential-like properties, namely: $f(x)$ is infinitely differentiable; $f(x)$ and all its derivatives are monotonic; $f(x)$ and all its derivatives ...
1
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1answer
10 views

Find 3 fixed points of function with 2 arguments

I'm looking for a way to determine the fixed points of a function with 2 arguments. The specific function is the following: $F:\bigl( \begin{smallmatrix} x \\ y \end{smallmatrix} \bigr) \mapsto ...
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2answers
30 views

Why can't a strictly injective function have a right inverse?

let $A = \{a \in A\}$ and $B = \{b \in B\}$. Let $f$ be a strictly surjective map $f: A \to B$ meaning for every $b$ in $f$'s codomain there must exist some $a$ in $f$'s domain. $f$ is surjective if ...
0
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0answers
28 views

Modifying permutation function for inputs with equivalent ratios

I have the following function: $$f(a_1,a_2,\ldots,a_n) = \frac{(a_1 + a_2 + \cdots +a_n)!}{(a_1! a_2! \cdots a_n!)}$$ where $a_i\ge 0$ I need to modify this function such that $f(a_1,a_2,...,a_n) = ...
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2answers
30 views

if $f(x)=x^2$ and $g(x)=x\sin x+\cos x$ then number of points where $f(x)=g(x)$?

The question is if $f(x)=x^2$ and $g(x)=x\sin x+\cos x$ then number of points where $f(x)=g(x)$? My approach:- $$f(x)=g(x)\implies x^2=x\sin x+\cos x\implies x^2-x\sin x-\cos x=0$$ Let ...
-1
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2answers
41 views

How to calculate the number of non decreasing functions between two finite sets? [duplicate]

I want to know how to calculate number of non decreasing functions from one set to another set. Let $A=\{1,2,3,\ldots,10\}$ and $B=\{1,2,3,\ldots,25\}$ Please tell me an easy method to calculate the ...
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2answers
23 views

Clarification Needed Regarding $\sinh^{-1}(-3)$

As the definition of $\sinh^{-1}(x)$ goes : $\sinh^{-1}(x)=\ln\left(x+\sqrt{x^{2}+1}\right)$ So what I expect to get is $\sinh^{-1}(-3)=\ln\left(-3+\sqrt{10}\right)$ The value inside of the ...
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1answer
10 views

Parity of multivariable functions

Given the functions $$\psi(r,\theta) = r^\lambda f(\theta)$$ $$u(r,\theta) = r^{-1}\partial_\theta \psi = r^{\lambda-1} f'(\theta)$$ $$v(r,\theta) = -\partial_r\psi=-\lambda ...
1
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0answers
84 views

Is this (set-)difference of spaces of functions nonempty?

Let $g_y$ be a family of continuous functions on $[a,b]$, indexed by $y\in [0,1]$, such that $(x,y)\to g_y(x)$ is also continuous. Denote the set of all such families that additionally fulfill ...
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2answers
17 views

Difference between Increasing sequence of functions and Sequence of increasing functions

Suppose we define a sequence of functions $\{f_n \}_{n \in \mathbb{N}}$. I am confused in the following terminologies regarding this sequence of functions - 1) Increasing sequence of functions 2) ...
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3answers
519 views

Different arrows in set theory: $\rightarrow$ and $\mapsto$ [duplicate]

Can someone explain the difference between symbols: $\rightarrow$ and $\mapsto$ Thanks.
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1answer
35 views

Evaluating an implicit function

Consider the functions defined implicitly by $y^3 - 3y + x = 0$. If $x \in (-\infty,-2) \cup (2, \infty), y=f(x).$ If $x \in (-2,2), y= g(x)$. Also, $g(0)=0$. Then, If $f(-10 \sqrt{2}) = 2 ...
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3answers
43 views

Let $f(x) = x^{2}$ for all $x \in \Bbb R$. Show that $f[\Bbb Q] \subset \Bbb Q$

Let $f(x) = x^{2}$ for all $x \in \Bbb R$. Show that $f[\Bbb Q] \subset \Bbb Q$ We know that $f[\Bbb Q]$ is the set of all values that $f$ takes on given points in $\Bbb Q$, i.e. $f[\Bbb Q] = ...
0
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0answers
38 views

A unbounded exponential function

Could you tell me how to show that the function $f(y)=xy-e^x$ is unbounded if $y <0$, where $x,y \in \mathbb{R}$?
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1answer
49 views

Show that each composite function $f_i \circ f_j$ is one of the given functions

I'm just going through the problems that I got wrong on my discrete math exam, and I was not sure how to do this one. How would I go about making this chart? The chart has $f_1, \dots, f_5$ going ...
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1answer
65 views

Looking for function $f$ such that $f'<0$ and $(xf)'>0$

I'm looking for a function $f(x)$ with the following properties for $x\ge 0$: $$0\le f(x)\le 1$$ $$f(0)=1$$ $$f'(x)\le 0$$ $$f(x)+xf'(x)\ge 0$$ $$\lim_{x\to\infty} xf(x)=L$$ where $L$ is a positive ...
0
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0answers
34 views

Proving 1:1 and onto of 2 variable function

Let $\ f: \mathbb Q^2\to\mathbb Q^2$ be defined by $\ f(x,y) = (x-3y , x + 3y- 1)$ Prove the function is 1:1, onto, find the inverse and $\ f \circ f $. I was able to prove 1:1 by showing $\ ...
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1answer
78 views

Proving that the $\sqrt 2$ exists with Pythagoras is valid. [closed]

My attempt is applying IVT for the continuous function $x^2-2$ on $[1,2]$. But I wonder that: Does $1,1,\sqrt 2$ triangle prove that the $\sqrt 2$ exists?
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1answer
35 views

True/False: Real Analysis: Series of Functions

For the following statements, we must either prove them true or find a counterexample. 1) If $\sum_{n=1}^{\infty}g_n$ converges uniformly, then $(g_n)$ converges uniformly to $0$. Thoughts: I think ...
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1answer
19 views

Can a sequence of functions with bounded derivative converge to something unbounded?

Suppose we have a sequence of continuous functions $h_n$ on the interval $[0,1]$ with uniformly bounded Lipschitz norm, $\sup_{x,y } \frac{|h_n(x)-h_n(y)|}{|x-y|}$. Let $h_n$ converge to some function ...
0
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0answers
30 views

Prove that either $f(-a_n)>f(0)$ or $f(a_1-a_n)>f(0)$.

I have a finite sequence of positive numbers $(a_i)_{i=1}^n$ and a function $f:[-a_n,a_1-a_n]\to\mathbb{R}$ such that ...
1
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1answer
28 views

How to prove a function of ordered pairs is injective

$f: (\mathbb{Z}^+{\times}\mathbb{Z}^+) \to \mathbb{Z}^+$ where $f(a,b)=2^a3^b$ How would I show that $f$ is injective?
3
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1answer
32 views

How many one to one functions?

how many one to one functions for $$p : (4, 5, . . . , 8) \to (4, 5, . . . , 8)$$ if: $p(6) = 6$. What about $p(6) \neq 6$? I thought 1 was 4! and 2 was $4*4!$, but that doesn't seem to be right.
0
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1answer
11 views

Construct a Sinusoidal Equation for an Irregular Period

I would like to be able to construct a sinusoidal function of limited domain given a set of real roots, assuming that the function is graphically centered on $y=0$. I expect that this would ...
0
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4answers
70 views

What is a representation?

I know the definition is given as follows: A map $p: G \rightarrow GL(V)$ such that $p(g_1g_2)=p(g_1)p(g_2)$ but I still do not really understand what this means Can someone help me gain some ...
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1answer
16 views

Tests for my “LineGraph-from-AdjacencyMatrix” function

I think I found a way to generate adjacency matirices $L$ of line graphs from the adjacence matrices $A$ of graphs. Now I want to test my function $L=f(A)$. When $A$ is the cube, $L=f(A)$ already has ...
1
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1answer
42 views

Evaluate $\int_{\partial \mathbb{B}(-i,3)}\frac{\sin(z)}{(z-3)^3}\, \mathrm{\mathop{d}}z$ using Cauchy's Integral Formula

I would like to evaluate $$\int_{\partial \mathbb{B}(-i,3)}\frac{\sin(z)}{(z-3)^3}\mathrm{\mathop{d}}z$$ using Cauchy's Integral Formula. Notation: We denote $\partial \mathbb{B}(a, R)$ to be the ...
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1answer
26 views

A question regarding critical points

We have the function $f(x)=\frac{1}{(x-5)}$. I want to find the critical points so I differentiated it and I got $f'(x)=-\frac{1}{(x-5)^2}$. Now I want to find the critical points so I have to find ...
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1answer
54 views

If $f$ is continuous real function such that $f(11) = 10$ and $\forall x, f(x)f(f(x)) = 1$ then find $f(9)$ [closed]

If $f$ is continuous real function such that $f(11) = 10$ and $\forall x, f(x)f(f(x)) = 1$, then $f(9) =$ ?
0
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1answer
19 views

Function for a sinusoidal camel plot?

Is this kind of function possible? If so, what would the equation look like? I can get the camel like notch with a an exponent in a sin, such as sin(x^2), but I only get one notch.
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1answer
48 views

What is the period of a fuction which satisfies the condition $f(a-x)=f(a+x)$?

What is the period of a function which satisfies the condition $f(a-x)=f(a+x)$ where a is any positive integer? I tried substituting $x$ with $x-a$ but that does not seems to help me a lot. I ended ...
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2answers
33 views

Value of $f'(0)$ if $f(1/x)=0,\ \forall x \in \mathbb{N}$?

If there exists a continuous and differentiable $(\forall x \in \mathbb{R} )$ function $f:\mathbb{R} \rightarrow \mathbb{R}$ such that $f\left ( \frac{1}{n} \right )=0$ for every integer n not ...
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4answers
31 views

Injective function for a given 'a'

If a function is given whose domain and codomain are all real numbers such that $$f(x) = x^3 + ax^2 + 3x + 100$$ then we have to find the value of $a$ for which the function is injective . For that ...
2
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1answer
46 views

what does linearly independent in C[0, 1] mean?

This is a question from my textbook I'm not quite sure what C[0, 1] mean, I tried to google the similar question and found that $C[0,1]$ usually denotes the collection of continuous functions $f: ...
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1answer
35 views

Is ${\{1\}}^\omega$ isomorphic to $\mathbb{Z}_+$?

Here is a proof that countably infinite product of countable sets is not (always/never?) countable. My question is : what if $X={\{1\}}$ ? i.e. Is $g:\mathbb{Z}_+\to X^\omega$ surjective (or even ...
0
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1answer
32 views

find monotonic, increasing function going exactly throught set of points

This questions stems from a problem that I encountered while writing a program. This problem was identical to this one. What I did on the spot was described as a "trick" with binary search over ...
2
votes
2answers
63 views

All-Russian Olympiad question (sum of symmetrical functions)

(All-Russian Olympiad, $1995$, $11^{th}$ Graders, Final Round) Prove that every real function, defined on all of $\mathbb R$, can be represented as a sum of two functions each of which has a vertical ...
2
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1answer
35 views

Second-order derivative with respect to a function of two variables.

I have a surface defined as a radius vector in spherical coordinates: $$r = r (\theta, \psi).$$ In Cartesian coordinates, the projections are calculated as follows: $$\begin{align} r_x &= r \sin ...
1
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1answer
27 views

Find the solutions for $a$, for which the function $f(x) $is non-decreasing

I came across an example which asks to find all the values of $a$, for which the function $f:\mathbb{R}\rightarrow \mathbb{R}: x\mapsto e^x(ax^2+1)$ is non decreasing. I derived it, $f'(x) = e^x(ax^2 ...
2
votes
0answers
18 views

How to prove $|n-m|\lt d\implies |f(n)-f(m)|\lt e$ for the following condition?

How to prove $\exists n\in\Bbb{N},\forall e\in \Bbb{R^+},\exists d\in\Bbb{R^+},\forall m\in\Bbb{N}, |n-m|\lt d\implies |f(n)-f(m)|\lt e$ if $f(x)=\lfloor \frac{n}{3}\rfloor$. I don't know how to ...