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Questions tagged [functions]

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

1
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5answers
28 views

Proving that a function is not injective.

The proof asks for a function $f: \mathbb{R} \rightarrow \mathbb{R}$ given by $$f(x)=\frac{1}{x^2+1}$$ prove that it is neither injective nor surjective. My thoughts are to approach this using a ...
0
votes
0answers
36 views

Prove that a function is linear

Let $U \subseteq \mathbb R^k$ be an connected open set, and let $f:U\to\mathbb R$. Suppose that $\forall p\in U$, there exists a closed set $C$ with non-empty interior such that $p\in C$ and $f(x)=...
1
vote
1answer
39 views

Characterize all functions $f$ such that $f(g(x))=f(x)$ if and only if $g(x)=x$

While working on a seemingly-unrelated research question, I've stumbled upon the condition I mentioned in the title. Is there a name for the class of functions $f$ such that $f(g(x))=f(x)\...
4
votes
3answers
39 views

Constructing an arithmetic progression so that $\sum_{i=1}^n f(x_i) =0$

Let $f: \mathbb{R} \to \mathbb{R} $ be a continuous function so that $ \exists a, b \in \mathbb{R} $ with $f(a) f(b) <0$. Prove that $\forall n>2 \exists$ an arithmetic progression $x_1<x_2&...
1
vote
1answer
24 views

Questions on the proof that $f$ below is $f\in C^\infty(\mathbb{R})$

Given the function, $$f(x)=\begin{cases}\exp\left(-\frac 1 x\right)&x>0\\ 0&x\leq 0,\end{cases}$$ it is a common exercise to show that $f\in C^\infty(\mathbb{R})$. An attempt to this ...
0
votes
0answers
17 views

Scalar function from $XYZ$ coordinates

As part of the project, I have to calculate the Gaussian, mean, max and min principal curvatures. I understand that I need to use partial derivatives to get them and I understand how to do that but I ...
1
vote
0answers
26 views

Show that a function is strictly convex, differentiable, continuous

Consider the vector $\theta\equiv (\theta_1,...,\Theta_M)$. Let $\Theta\subseteq \mathbb{R}^M$. Consider the random vector $\epsilon\equiv (\epsilon_1,..., \epsilon_M)$. (A1) We assume that the ...
0
votes
0answers
14 views

When is reversing composition order monotonous?

Let $f_a(x)=x+\exp(-ax)/a$ be a function, defined on $R_+$ for $a$ positive. Fix $a$ and $b$ positive, can we show that the sign of $f_a\circ f_b - f_b\circ f_a$ does not change on $R_+$, with $\circ$ ...
0
votes
0answers
18 views

Prove that the limit of a continuity function equals the limit of the corresponding sequence

following question: Let $U$ be a monotonically increasing function. If $\lim\limits_{n \to \infty} \frac{U([t]x-b_{[t]})}{a_{[t]}} = D(x)$ holds for every continuity point $x$ of D with $[t]$ being ...
0
votes
1answer
11 views

convert iterated function to continuous function

Let's say we have a function: $$v(t_n) = a\cdot(t-t_{n-1}) + v(t_{n-1}) \cdot d^{t-t_{n-1}}$$ Where $v$ = veolcity $t$ = time $a$ = acceleration $d$ = friction or damping What is basically does is ...
-1
votes
2answers
27 views

Intervals of a derivative

I'm trying to find the intervals of the derivative of a function that I found on my book but I'm having some troubles understanding it, so I thought of trying to find some help here. The function is $...
2
votes
3answers
69 views

How to find the range of a composite function?

I have been stuck at this question: I have $$f(x)=\cos(\pi \cdot x)$$$$g(x)=\frac{7\cdot x}{6}$$ and $$h(x)=f(g(x))$$ and i am asked to compute the range for $h(x)$. My solution: $$h(x)=\cos(\pi \...
1
vote
2answers
37 views

Find the closure of $C=\{f\in C([0,1]):f(0)=0\}$

Let $C([0,1])$ be the space of all real valued continuous functions $f:[0,1]\to \mathbb{R}$. Take the norm $$||f||=\left(\int_0^1 |f(x)|^2\right)^{1/2}$$ and the subspace $$C=\{f\in C([0,1]):f(...
0
votes
0answers
7 views

projection functions class closed by primitive recursion and composition?

I've stumbled upon the following exercise and I am having some doubts with the solution: Let $\mathcal{C}=\{{u_i}^n:1\leq i \leq n\}$ be the class of all the projection functions. Decide if the ...
0
votes
1answer
21 views

Bijectiveness in a neighborhood of $(0,\pi/2)$

Question: Define $g: \mathbb{R^2}\to \mathbb{R^2}$ by $g(x,y)=(y\cos x,(x+y)\sin y)$. Show that g maps a neighborhood of $(0,\pi/2)$ bijectively to a neighborhood of $(\pi/2,\pi/2)$. What I did/...
0
votes
1answer
23 views

Can the domain also be referred to as the element of the set?

I understand that the domain of a function is the set of all input values, but in some textbooks, the domain is an element of that set of input values. How can a set also be a subset of that set?. ...
0
votes
0answers
17 views

Rewrite conditional distribution that is an inequality as a function

How do I rewrite a conditional distribution that is an inequality as a function of another variable? The problem I have been posed is the following: Given $F_u(y) = \mathbb{P}[A-u \le y | A > u]$...
-2
votes
2answers
42 views

Draw a graph of function $y=\sqrt {x+3-2\sqrt{x+2}}+x$ [on hold]

$$y=f(x)=\sqrt{x+3-2\sqrt{x+2}}-x$$ How it is possible to simplify this function to draw it on a plane?
0
votes
1answer
30 views

How do I calculate area?

How do I calculate the area of this: $D=\{ (x,y)\mid 0 \le x \le 1, x^2 \le y \le x^2+2 \}$ ${A}=\iint_D \, \textrm{d}A.$ Don't know what is the right answer but I have get that the area is 6. Is ...
0
votes
0answers
23 views

number of functions satisfy given equations

Number of possible continuous function $f:[0,1]\rightarrow (0,\infty)$ for which $\displaystyle \int^{1}_{0}x^{k}f(x)dx=a^k $ for $k=0,1,2$ and $a\in\mathbb{R}$ what i try $\displaystyle \int^{1}_{...
1
vote
1answer
32 views

Does a set of all real valued functions form group under componentwise multiplication?

I think I have verified the first two axioms: closure, associativity. For existence of identity, I said that $\forall$ $f$ $\in$ $\mathbb R^\mathbb R$, there is a function $e(x) = 1$ $\in$ $\mathbb R^\...
1
vote
1answer
47 views

How to solve this type of differential equations using power series?

$$ y''(x) + f(x)\cdot y = 0 $$ I'm struggling with this differential equation and i've stuck every time.
0
votes
2answers
29 views

Functions on $\mathbb{R}$ symmetric about a point $x_0 \in \mathbb{R}$

We can say that function $f(x), x \in \mathbb{R}$ is symmetric about a point $x = x_0$ if $f(x_0 - x) = f(x_0 + x)$ for all $x$ for which $f$ is defined. Now let's try to find a function which is ...
1
vote
1answer
35 views

Finding support of a function

Let $X$ be a set and $f$ be a function from $X$ to $\{0, 1\}$, the field with two elements. The support of $f$ is the set $f ^ {-1}$ (1), which we denote by $\DeclareMathOperator{\supp}{supp}\supp(f)$...
3
votes
1answer
35 views

$f(x) = \begin{cases} 0 & x\in( \mathbb{Q}\cap [0,1])^c \\ \frac{1}{q} & x=\frac{p}{q} \in \mathbb{Q} \cap [0,1], (p,q)=1 \end{cases}.$

let $f$ be the function defined by $$f(x) = \begin{cases} 0 & x\in( \mathbb{Q}\cap [0,1])^c \\ \frac{1}{q} & x=\frac{p}{q} \in \mathbb{Q} \cap [0,1], (p,q)=1 \end{cases}.$$ Prove ...
-1
votes
1answer
41 views

A x B as a function

So I am trying to prove if AxB is a function where: 𝑓: 𝐴 → 𝐵 𝐴 = {1, 2, 3, 4} 𝐵 = {2, 4, 5} I am able to do this with for example {(1, 4), (2, 2), (3, 5), (4, 4)} As far as I understand one ...
0
votes
0answers
28 views

How to construct a function with these hypotheses?

I want to construct a function $f:[0,1]×[0,1]\rightarrow [0,1]$ such that $f(0,t)=t$ $f(1,t)=2t-1$ $ \forall$ $ t\geq \frac{1}{2}$ (Edited) $f(s,t)=0$ $ \forall $ $0 \leq t \leq \frac{s}{2}$ (Edited) ...
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0answers
11 views

Function application question [on hold]

https://imgur.com/a/m3erggc Question 2c. Anyone know if my answer is correct?
1
vote
1answer
44 views

Is the function $f: \mathbb{Q}\to \mathbb{R}$ given by $f(r)=\frac{p}{10^q}$ one-one and onto?

Is the function $f: \mathbb{Q}\to \mathbb{R}$ given by $f(0)=0, ~$$f(r)=\frac{p}{10^q}$ where $p\in \mathbb{Z}$ and $q\in \mathbb{N}$, $(p,q)=1$, one-one and onto? Since, for $\sqrt{2}\in \mathbb{R}$ ...
2
votes
1answer
26 views

Find volume of solid using shell method

Use the method of cylindrical shells to find the volume of the solid obtained by rotating the region bounded by the curves $y=x^2$, $y=0$, $x=1$, and $x=2$ about the line $x=4$. Here is what I ...
5
votes
2answers
76 views

Find $a,b$ at $f(x)=\frac{x^2+x-12}{x^2-ax+b}$

An High school question: Given :$$f(x)=\frac{x^2+x-12}{x^2-ax+b}$$ it's given that $x=3$ is a vertical asymptote find $a$ and $b$. I tried: Since $x=3$ is a vertical asymptote then $3^2-3a+b=0$,...
0
votes
3answers
64 views

How to prove that $f(x) =1/x$ is unbounded in $(0,1)$?

Let $f(x) = 1/x$ for all $x\in (0,1)$ By assuming $f(x)$ is bounded or in any other way but without using limits I assumed that $f$ is bounded above, then there is $M>0$ s.t $f(x) \leq M$ for all ...
2
votes
1answer
42 views

Properties of a step function

Consider the step function $\Delta: \mathbb{R}\rightarrow [0,1]$ $$ \Delta(x;\lambda,\mu)\equiv \sum_{j=1}^J \lambda_j\times 1\{\mu_j\leq x\} $$ where $\lambda\equiv (\lambda_1,...,\lambda_J)$ is a ...
-3
votes
1answer
34 views

Advanced Functions Chapter 5 [on hold]

The turntable for some vinyl records spins with an angular velocity of 45rpm. Determine the exact angular velocity of the turntable in the radian per second.
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votes
2answers
58 views

Advanced Functions Chapter 4 [on hold]

Use a counterexample to show $\cos(3x)=3\cos x-4\cos^3x$ is not an identity.
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votes
1answer
36 views

Advanced Functions Chapter 4/5 [on hold]

The pendulum in a grandfather clock is exactly 1 meter long. If the pendulum swings through an angle of 75 degrees, determine the exact distance the bottom of the pendulum travels in one swing.
0
votes
2answers
29 views

Determine the domain, co-domain and range - again

I am trying to determine the domain, co-domain and range of the following function A function assigns to each integer, the square of that integer multiplied on 4. What I think it is, is: f: X→√ℤ*4 ...
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votes
0answers
34 views

How do you perform a function?

The author of my additional maths textbook was trying to explain the technique of differentiating composite functions, using a particular function: $y=1/(3x-4)^2$. He then wrote, and I quote "to find $...
6
votes
4answers
195 views

Why does any function get thinner as $x$ is multiplied by a constant?

Example: $$\cos(x)$$ $$\cos(8x)$$ "Thinner" might not be the correct term. But I just want to know why does changing $x$ to $8x$ make it look like that?
0
votes
1answer
50 views

In any triangle is sinA+sinB+sinC=(3Root3)/2 always

Well I came with an interesting proof. But I just want to verify it From here we will get sinA+sinB+sinC=<(3root3)/2 and from this I get sinA+sinB+sinC>= (3root3)/2 Now the equation to be ...
1
vote
0answers
35 views

Why is this injection from $\mathbb{N^N}$ to $\mathbb{N}$ wrong? [duplicate]

I was looking to generate an injection from $\mathbb{N^N}$ to $\mathbb{R}$, and this is what i created: $$ \begin{align*} g: \mathbb{N^N} \hookrightarrow \mathbb{R} \\ f \mapsto p_1^{f(0)}p_2^{f(1)}...
0
votes
1answer
22 views

Find the interval of $k$

Given $f(x)=\frac{e^x}{x^2}-k(\frac{2}{x}+\ln(x))$, $k$ is a constant. If $f(x)$ has two extreme points at the interval of $(0,2)$. Then find the interval of $k$. My approaching is: I assume the x-...
1
vote
1answer
66 views

Show that $x \in W_{x}$ is undecidable

The Cutland's book called Computability has a theorem whose proof i don't understand and i have developed another simpler proof. Could you tell me if this proof is correct? DEFINITION 1: $\phi_{x}, x ...
-2
votes
3answers
72 views

Finding Integer $x$ Values that $f(x)$ Is a Square Number [on hold]

So I was wondering, for $$f(x)=x^2+12x+3$$ is there a way to identify the integer $x$ values that $f(x)$ is a square? But, I'm looking for ways that do not include factoring, but general algorithms ...
0
votes
1answer
33 views

How do we qualify the successor function and other maps of large classes?

When I took a course in set theory last year, I remember being rather intrigued by the definition of the successor function/operation $ S(x) = x \cup \lbrace x \rbrace $ , given that it is not a "...
2
votes
3answers
38 views

when $\sqrt{y}$ and $\sqrt{x}$ are defined, is $\sqrt{y}$ = $\sqrt{x}$ a function?

when $\sqrt{y}$ and $\sqrt{x}$ are defined, is $\sqrt{y}$ = $\sqrt{x}$ a function? for (x,y) in the reals. I think I'm thinking to hard about what the graph will look like
0
votes
1answer
59 views

Which $f$ is not a function from $\mathbb{R}$ to $\mathbb{R}$

I am having a hard time understanding how to begin solving this task. Am I suppused to solve the function to solve this task? And if so, how can I determine if $x$ in $f(x)$ is a real number to ...
1
vote
2answers
34 views

What is the breakpoint of a piecewise function?

For $$ f(x)=\left\{\begin{matrix} 0, & x\leq -1\\ \sqrt{1-x^2},& -1 < x < 1 \\ x, & x\geq 1 \end{matrix}\right. $$ My book says that the breakpoints are x = -1 and x = 1. How ...
0
votes
1answer
38 views

Determine the domain, co-domain and range [on hold]

I am trying to determine the domain, co-domain and range of the following function A function assigns to each bit string, the number of zeroes in that bit string. I imagine that this will be that "...
0
votes
3answers
33 views

Is it a function when one output isn't used?

I have a very basic question. From what I have learned it is not a function when one of the outputs from a ordered set isn't used. Is this correct? I am trying to determine if this relation is a ...