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

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1answer
49 views

Does every quartic polynomial of the form $(x+a)(x+b)(x+c)(x+d)$ where $a \neq b \neq c \neq d \neq 0$ have a distinct local and absolute minimum?

To me it seems like yes because it's composed of 4 linear factors so it would have four roots. Also it's not negative fourth degree, so therefore has two convex lumps in the function. Perhaps ...
0
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3answers
53 views

Given two real functions, $f$ and $g$, if $|f(x)|<1$ then $|g(f(x))|<g(1)$? Why?

It seems trivial for a certain $x$ but can we say it for all x?
-2
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1answer
22 views

Discrete Mathematics question regarding functions. [on hold]

Let $S = \{s_1,s_2,...,s_n\}$. How many functions are there with domain $S$ and target Z2? Of those functions, how many are one-to-one? How many are onto?
1
vote
1answer
79 views

Let $f$ be injective and discontinuous at some point $c$. Can its inverse be continuous?

$f$ is injective at an interval $[a,b]$, but discontinuous at some point $c$ in the same interval. I need to prove that its inverse is continuous at that interval. Should I consider what is the ...
0
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2answers
40 views

Let f be continuous. By EVT there exists a c such that f(c)=supx f(x). Show that f is not injective.

I am given a continuous function f in an interval [a,b]. To show that f is not injective, should I consider the definition of the extreme value theorem? I am not sure how to show that it is not one ...
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3answers
31 views

$\int\limits_0^1 {\left( {1 - 2{x^2}} \right)f\left( x \right)dx}<0$, when $f$:convex and differentiable with $f(0)=0$

Let $f:[0,1]\to\mathbb{R}$ be a differentiable function that is convex and $f(0)=0$. Prove that: $\int\limits_0^1 {\left( {1 - 2{x^2}} \right)f\left( x \right)dx}<0$. I thought that since ...
0
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1answer
18 views

How to find $x$-intercept on TI-83/TI-84 calculator without having to set the bounds for each intercept?

Finding $y$-intercepts is very easy on a TI-83/TI-84 calculator. All you have to do is graph the function and use 2nd $\to$ ...
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0answers
14 views

Solving $f(x) \leq 10 f(kx) + 10kg(x)$ for $f, g$ nonnegative on $(0, 1]$

Suppose we are given two nonnegative functions $f$ and $g$ on $(0,1]$ that satisfy $f(x) \leq x^{-1/2}$ and $$f(x) \leq 10 f(kx) + 10kg(x)$$ for all $k$ sufficiently large. Is it possible to reduce ...
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0answers
16 views

Represetation of a smooth function in the neighborhood of its zero-set

Consider $k$ smooth functions $g_i(x)$, $x\in \mathbb{R}^n$, $k<n$. The set $G$ is defined as $G=\{x\in \mathbb{R}^n|g_i(x)=0, i=1,...,k\}$. We also assume that the Jacobi matrix $\frac{Dg}{Dx}$ is ...
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1answer
40 views

CALCULUS: Sketching a function by given conditions [on hold]

Pls help. I'm currently on a struggle with this calculus problem. Thanks in advance.
0
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1answer
36 views

Meaning of square root in this situation

I wasn't someone who really paid attention in math class but grew very fond of it throughout the time I was out of school. Square Root = A number that produces a specified quantity when multiplied by ...
1
vote
1answer
17 views

Product of real-valued functions on $\mathbb{R}^n$

Let $f,g:\mathbb{R}^n\to \mathbb{R}$. What is $fg$? Is it function $\mathbb{R}^n\to \mathbb{R}$ or $\mathbb{R}^{2n}\to \mathbb{R}$? For example, taking $n=2$ let $f(x_1,x_2)=x_1+x_2+x_1x_2$ and ...
1
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1answer
11 views

Continuous functions on compact group and uniformity

If $G$ is a compact abelian group and $f\in C(G)$. Then $\forall \epsilon >0$,there exists an open neighbourhood $U$ of $0\in G$, such that $\forall g\in G , \forall u_1,u_2\in U$, we have ...
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0answers
33 views

$f(x) = 2x \mod 1$ not equal to zero for all $x$?

If any number $\mod 1$ is zero, then how can $f(x) = 2x \mod 1$ be a Baker's map? For any $x\in \mathbb{R}$, shouldn't $f(x)=0$?
1
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1answer
31 views

Prove that $ϕ ◦ ϕ = ϕ^2 = Id_{\Bbb C}$ (the identity map on C) if and only if $e^{iθ} \bar c + c = 0$.

Consider the isometry $ϕ : \Bbb C → \Bbb C$ with equation $ϕ(z) = e^ {iθ} \bar z + c$ where $θ ∈ \Bbb R$ and $c ∈ \Bbb C$. Prove that $ϕ ◦ ϕ = ϕ^2 = Id_{\Bbb C}$ (the identity map on C) if and only if ...
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1answer
33 views

Prove closed disc $D^n$ is homeomorphic to the cone $CS^{n-1}$

I need to find a continuous surjective map from $D^n$ to $CS^{n-1}$. For 2 dimensions, we can use $$f: S^1 \times I /S^{1} \times \{1\} \rightarrow D^2$$ with $f(\theta,t) = (1-t)e^{i \theta}$ ...
1
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2answers
25 views

Constancy of an integral function

Fix some $\ell\in\mathbb{R}^+$. Say that $f:\mathbb{R}^2\to\mathbb{R}_{\geq0}$ and $\mu:\mathbb{R}\to\mathbb{R}^+$ are functions satisfying the following: $f$ and $\mu$ are continuous. $f$ is ...
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3answers
46 views

Find the area of the region described by $|5x|+|6y| \le 30 $

Find the area of the region described by $|5x|+|6y| \le 30 $ (where $|z|$ denotes the absolute value of $z$). My effort Imagining a number line and interpreting the problem as the request to ...
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1answer
32 views

Is torus w. disc removed homotopic to klein bottle w. disc removed?

I know that homeomorhic spaces are homotopic, but am not sure if this applies, since I think they are not homeomorphic due to orientability. I know f and g are homotopic if they represent: ...
2
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2answers
47 views

Find $f$, when $f(1)=f(2)=1$ and $f'(x)\leq (x^2-x-1) e^x, \forall x\in [1,2]$

Let $f:[1,2]\to\mathbb{R}$ be a differentiable function such that $f(1)=f(2)=1$ and $f'(x)\leq (x^2-x-1) e^x, \forall x\in [1,2]$. Find $f(x)$. I am pretty sure that from things I have tried ...
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0answers
12 views

Determining function of Graph

Looking through some notes on numerical computation I came across the following graph: I know this is a longshot, but I'm not incredibly mathsy and would like to know what kind of a function this ...
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2answers
60 views

A weaker than “zero derivative” condition implies that a function is constant?

Let $f:(a,b)\to R$ be a continuous function such that $\limsup\limits_{n\to \infty}\frac{f(x_n)-f(x_0)}{|x_n-x_0|}\leq 0$ for every $x_0\in(a,b)$ and sequence $x_n$ converging to $x_0$ such that ...
2
votes
3answers
43 views

Taylor expansion of $\cos{x}$

I found a pdf file on the internet which gives you known expansions of Taylor's. There is something I cant understand : Why is the remainder of $\cos x$ is written like this? $$\frac{\cos ...
3
votes
3answers
33 views

Determine whether $e^x(1-e^x)\leqslant (1/4)$ for $x\lt 0$ is true or false?

The statement is $e^x(1-e^x)\leqslant (1/4)$ for $x\lt 0$ I think the above statement is true by calculating approximate value at some points. But how to prove it properly?
0
votes
1answer
19 views

Discrete Math Compositions

I am having trouble with these compositions. $$T = \{(a,a), (a,b), (b,c), (b,d), (c,d), (d,a), (d,b)\}$$ $$U = \{(a,a), (a,d), (b,c), (b,d), (c,a), (d,d)\}$$ I need to find $T \circ T$, $U \circ T$, ...
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1answer
52 views

Finding the domain of $\frac{1}{x}|x^2 - 1|$ [on hold]

What is the domain of this function $F(x)=\frac{1}{x}|x^2 - 1|$ Can someone please tell me how to find it ?
1
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2answers
25 views

Why coefficients have to be proportional for two quadratic functions to have the same roots?

We have the next two quadratic functions: $ ax^2 + bx + c = 0 $ $ mx^2 + nx + p = 0 $ If $ a/m = b/n = c/p $ then they have the same roots. What is the intuition behind this statement?
5
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0answers
33 views

Find the inverse of the following piecewise defined function

Find the inverse of $f$ if $f(x)=$ $$ \begin{cases} \sqrt{2-x}, &\text{for $x<0$}\\ 1-x^2, &\text{for $x \ge 0$} \\ \end{cases} $$ My effort For $y=\sqrt{2-x}$ ,we find ...
0
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0answers
18 views

Need function/formular for time manipulation book

this might be a rather unusual question, however: I'm working on a book idea and am looking for a mathematical function or some formular etc. Here is what I have in mind, so you might get a better ...
1
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2answers
65 views

Why is the function continuous at a point which gives the case 0/0?

I have this function : $f(x) = \frac{6x^2+18x+12}{x^2-4}$, the domain is R. How come its graph is continuous at $x = -2$? I know it can be simplified to $\frac{6(x+1)}{x-2}$ ( firstly $f(x) = ...
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0answers
20 views

Calculus: Proving Continuous Function by Intermediate Value Theorem [duplicate]

Prove step by step: Let $f(x)$ be a continuous function from the closed interval $[a, b]$. Use the Intermediate Value Theorem to show that $f(x)$ has a fixed point, that is, there is a point $x \in ...
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2answers
30 views

What is the instantaneous rate of change in the real world?

I can't grasp this concept of an instantaneous change of rate. How could a point on a function graph have a rate of change in the first place? In this moment I just know that it is named the ...
2
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0answers
41 views

How to prove that the following function has a unique mode?

I am trying to prove that the function $$f(\alpha)=n\ln \alpha-n\ln\Big(\sum_{i=1}^{n}t_i^\alpha+\int_{a}^{b}x^{\alpha+\beta-1}e^{-\lambda x^\beta}\,dx\Big)+(\alpha-1)\sum_{i=1}^{n}\ln t_i,$$ where ...
1
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2answers
43 views

Uniqueness of log function with relaxed conditions?

Question If: $$f(a) + f(b) = f(ab)$$ $$ f(1) = 0 $$ $$ a<b \implies f(a) < f(b) \forall a,b \in N $$ where $N$ is the set of natural numbers. Prove or disprove $f$ must be the $\log$ ...
0
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0answers
49 views

Prove that the following function is convex?

I am trying to prove that the function $$g(\alpha)=\ln\Big(\sum_{i=1}^{n}t_i^\alpha+A(\alpha)\Big) ~~t_i, \alpha>0,$$ where $A(\alpha)=\int_{a}^{b}x^{\alpha+\beta-1}e^{-\lambda x^\beta}\,dx$,is ...
2
votes
1answer
52 views

In which cases are $(f\circ g)(x) = (g\circ f)(x)$?

I have found three cases: 1) If $f$ and $g$ are the same function. 2) If $f$ and $g$ are mutually inverse. 3) If both are polynomials of degree $1$ Maybe there are more.
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0answers
33 views

how to get the solution?

I take the next exercise from Apostol's book. I thought for a while , but nothing. Then I read the solution. I try to think: how did he get it , but nothing. I want to know how to get to the solution. ...
1
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1answer
24 views

Function whose $n$-th derivative at $x=0$ is $n^3$ / evaluating the power series $\sum_{n=1}^\infty n^3\frac{x^n}{n!}$

I have troubles proving that the power series $\sum_{n=1}^\infty n^3\frac{x^n}{n!}$ represents the function $f(x)=e^x(x^3+3x^2+x)$. My idea was using the identity theorem for power series and the ...
4
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2answers
36 views

Finding ranks and nullities of linear maps

I am confused about ranks, nullities and bases of the kernel. From what I understand the rank is the dimension of a vector space generated by a matrix. How would I do the following examples? Find ...
2
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2answers
25 views

Vector-Valued Functions and Continuity

Why is it that when a vector-valued function $r(t)$ is continuous at some time $t$ then $\|r(t)\|$ is also continuous at that time $t$, but the converse is not true? That if $\|r(t)\|$ is continuous ...
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2answers
36 views

Show that an inverse of a bijective linear map is a linear map.

So I've got a bijection. It clearly has an inverse, but how exactly do I prove that the inverse is a linear map as well? enter image description here
3
votes
2answers
51 views
+200

Codomains and the definition of a function

A function $f$ is defined as a set of ordered pairs $(x, y)$ such that $(x,b), (x,c) \in f \Longrightarrow b=c$. Since $y$ is determined uniquely by $x$, it is customarily denoted $f(x)$. One ...
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2answers
50 views

Find the inverse $\dfrac{x}{\|x\|}$ in $\mathbb{R^2}$

I wish to find the inverse of $\dfrac{x}{\|x\|}$, where $x \in \mathbb{R}^2$ Let's do this. Let $$y_1 = \dfrac{x_1}{\sqrt{x_1^2+x_2^2}}$$ $$y_2 = \dfrac{x_2}{\sqrt{x_1^2+x_2^2}}$$ Then $$y_1 = ...
0
votes
1answer
21 views

How can I show the points of continuity of the following function

How can I show the points of continuity of the following function $$f(x) = \begin{cases} 2x, & \text{if $x \in \Bbb Q$} \\[2ex] x+3, & \text{if $x \in \Bbb I$ } \end{cases}$$ I am having ...
0
votes
2answers
17 views

Neural Network Sigmoid Problem

I currently try to create a three layer neural network and I want to use the backpropagation. My training data is: Input : Output 0 | 1 : 0 1 | 1 : 2 In many ...
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0answers
22 views

Function Limit & Continuity [on hold]

What is Function Limit & Continuity? I'm a little bit silly.Is there anyone to explain those terms precisely? Thanks in Advance...
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2answers
60 views

Help with continuity [on hold]

Could you please clarify these questions to me. Find all the numbers for which the given function is discontinuous. $F(x)=[x-1]$ I think the solution is $\Bbb Z$ numbers right ? $F(x)= ...
0
votes
1answer
15 views

If $f(r,t)=g(r),\,\forall r,t$ would that make $f$ and $g$ constant functions?

I know that if $f(r)=g(t),\,\forall r,t$ then $f(r)=g(t)=constant$, but If $f(r,t)=g(r),\,\forall r,t$, where now $f$ depends on $t$ would that lead to the same conclusion i.e. ...
-1
votes
0answers
39 views

Simplify $|a+sx|$ if $|x|=x$ and $|s|=1$. [on hold]

Simplify $|a+sx|$ if $|x|=x$ and $|s|=1$. Forgive me, this is probably really simple. Thanks. Edit: $a=v+sx$, where $v$ is a symmetric random variable with zero mean. So we can't say anything ...
0
votes
0answers
18 views

Is there a way to reverse engineer an already large number to make it smaller? [on hold]

Using the Ackermann function for example, it's quite easy to make massive numbers. My question is whether there's an existing algorithm that can take a large number and reverse engineer it to make an ...