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

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2
votes
2answers
31 views

Writing the integral of f(x) without any variables to find C.

I was given the equation $f(t) = 2t+3$ with an interval of [-3, 6], and I'm told to write a function for F(t) where $F(x) = \int_3^xf(t)dt$. Knowing integration I understand that the anti-derivative ...
2
votes
2answers
63 views

$F'(x) = f(x)$, but that's not working for me, help?

I'm working in an online class that's reviewing the usage of $F'(x)$, and I was just given the equation $F(x) = \int_x^1(\sqrt{1+t^2}\,dt$, what is $F'(1)$. Knowing that $F'(x) = f(x)$, I solved ...
0
votes
1answer
22 views

FUNCTIONS : Theoretical Doubt

I am currently learning calculus of one variables , and i have come across a symbol $$f(x,y).$$ Can anybody explain the meaning of this ? Thanks!
1
vote
2answers
32 views

Find $f(1)$ and $f'(1)$ of $\lim_{h\rightarrow\ 0}\frac{f(1+h)}{h} = 5$

Suppose the function, $f$, is differentiable at $x = 1$. $$\lim_{h\rightarrow\ 0}\frac{f(1+h)}{h} = 5$$ Find a) $f(1)$ and, b) $f'(1)$. I know b) (well at least I think it can) can be found by the ...
-1
votes
1answer
38 views

Inequality $\frac{4x-9}{3x^2+2} < y$

I need help with this equation: $\frac{4x-9}{3x^2+2} < y$ I need to solve for x: $x < ...$ The best i got so far was: $x(4-3xy)<2y+9$ Cant find a solution and would be very thankful for ...
-4
votes
0answers
19 views

Prove that $\left |f(x) \right |\leq \frac{x^2}{a}$ for every real number x. [closed]

For $a\geq 1$ is a real number and f:R→R is a function of simultaneously satisfying two conditions +, $(f(ax))^2\leq a^3x^2f(x)$ for every real number x +, f Blocked out in a certain neighborhood ...
0
votes
1answer
35 views

Equation gives a constant value of x?

The given question is : Find Maximum value of $f(x) +f(\frac{1}{x}) =\frac {1}{x}$ , $x \in$ domain of f I put $x=\frac {1}{x}$ to get $f(x) +f(\frac{1}{x}) =x$ Now from the above equations I get ...
0
votes
0answers
25 views

Find linear price function from a point

I have to do a simple managerial economics exercise. I know that a firm sells $Q=16000$ units at a price of $P=1672$. Moreover, I know that it will sell 18% more units if the firm decrease the ...
0
votes
3answers
44 views

Can't prove equation

Recently I have been working on an algorithm and haven't been completely able to continue because I've been stuck trying to create the formula of this curve. The curve has the following points: ...
0
votes
1answer
26 views

What is the domain of this function?

Let $f(x)$ be the following function: $$f(x)=\int_{0}^{2\pi}\dfrac{1}{\sqrt{1+x\cos t}}dt$$ How to find the domain of $f(x)$? I know that we should have $$1+x\cos t>0,$$ then I start by saying ...
1
vote
1answer
15 views

Describing a partition as a function

Let P be a partition such that P $=\{t_0,...,t_n\}$ over the interval $[a, b]$, then to refer to a point $i$ in the partition $P$, some would say $t_i$. So my question would be then, in this case ...
0
votes
0answers
26 views

Proving that these functionals are bounded, and finding their norms.

Proving that these functionals are bounded, and finding their norms. $$a.)f_1(x):c_o \to \mathbb R , f_1(x)=\sum_{n=1}^{\infty}\frac{x_n}{2^{n-1}} \\ b.)f_2(x):l_1 \to \mathbb R , ...
1
vote
1answer
68 views

$\lim_{x\to a}(f(x)+\frac{1}{|f(x)|})=0$. Find $\lim_{x\to a}f(x)$ [duplicate]

Let a function $f$ be defined in a hollow neighborhood of $a\in \mathbb{R}$, and suppose : $$\lim_{x\to a}\left(f(x)+\frac{1}{|f(x)|}\right)=0$$ Find $\lim\limits_{x\to a}f(x)$ and prove that this ...
-1
votes
1answer
60 views

$f$ is a constant function or not? [closed]

Let $f$ be a continuous function at $\mathbb R$ and determined $f: \mathbb R\rightarrow\mathbb R$. If $f(x)=f(x^2)$ for every real $x$, than $f$ is a constant function?
0
votes
2answers
27 views

right-cancellative property and surjectivity

I was trying to prove that if $f:X\to Y$ is a function (between sets $X$ and $Y$), then $f$ is surjective if and only if $f$ is right-cancellative: For all $g,h:Y\to Z$, if $g\circ f=h\circ f$, then ...
-2
votes
3answers
55 views

Prove that $f$ is a constant function. [closed]

Suppose that $f$ is a function determined $f:\mathbb{R}\rightarrow \mathbb{R}$, if $f(x)=f(2x)$ then is $f$ a constant function?
2
votes
1answer
133 views

A problem of olympiad. [closed]

This nice functional equation was proposed in the “VIII Olimpíada Iberoamericana de Matemáticas” held in Mexico (1993). Find all the functions $f:\mathbb N^* \to \mathbb N^*$ such that i) ...
4
votes
1answer
77 views

Does “the functions agree at infinity” mean anything?

I want a way to describe how two continuous functions $f,g \colon (X-x) \to Y$ might "share a limit" at the point $x$ when unfortunately neither of $\displaystyle \lim _{y \to x}f(x)$ or ...
0
votes
2answers
29 views

$f(x) = x^2 - \sin2x$ function, slope and degrees

I'm new here and sorry for my bad English, not my first language. Anyway, I have this function: $f(x) = x^2 - \sin2x\;,\;\;\left[-\frac\pi2< x <0\right]$ And I've been asked to find what is: ...
3
votes
1answer
33 views

Let $f(x)=x^5$. For $x_1>0$, let $p_1=(x_1,f(x_1))$.Draw a tangent at the point $p_1$

Let $f(x)=x^5$. For $x_1>0$, let $p_1=(x_1,f(x_1))$. Draw a tangent at the point $p_1$ and let it meet the graph again at point $p_2$. Then draw a tangent at $p_2$ and so on . Show that , the ratio ...
0
votes
0answers
9 views

ceiling joists storage expression

I have pre made truss in garage which has ceiling joist at $2 \times 4 \times 21$ feet, I want to sister a $2 \times 6 \times 21$ foot or $2 \times 8 \times 21$ next to $2 \times 4 \times 21$ how ...
1
vote
1answer
45 views

Estimate trigonometric functions with complex argument

I would like to prove the following estimates $\vert \sin(z)\vert\leq \sinh(s)$ and $\vert \cos(z)\vert\leq \cosh ( s )$ ,where $z\in D_s(0)\subset\mathbb{C}$ and $D_s(0)$ denotes the disc with ...
0
votes
2answers
74 views

What is $\operatorname{syt}$?

I came across the following definition of the set on this web page But what is $\operatorname{syt}$?
0
votes
1answer
23 views

Find approximation for size of population over time

Assume you start with a population of an objet of size $1$. Assume that a new objet of size $1$ is born at each date and that existing objects double in size in each period. Over time the sequence of ...
1
vote
1answer
42 views

Continuous and Inverse function

I need to prove that if $X$ is a subset of $\mathbb{R}^n$ and $Y$ is a subset of $\mathbb{R}^m$, and $X$ and $Y$ are closed and bounded, then if $f:X \rightarrow Y$ is continuous and has a inverse ...
1
vote
2answers
26 views

To show that for every continuous function there exists some other continuous function satisfying this conditions

Suppose that we start with some continuous function $f$ defined on $[a,b]$. Since it is continuous it is integrable so the number $\int_{a}^{b}f(x)dx$ exists. How to show (in an as elementary as ...
2
votes
3answers
66 views

FUNCTIONS : Theoretical doubt on functions 2

In the functional mathematics language , if i represent function by $$f$$ . What is the theoretical difference between$$f$$ and $$f(x)$$ ? Please provide a lucid explanation.Thanks.
1
vote
0answers
40 views

Support function of a set: when we can replace the supremum with the maximum?

Consider a set $K\subseteq \mathbb{R}^d$ (not necessarily closed and/or convex). Let $k:=(k_1,...,k_i,...,k_d)$ be a generic element of $K$. Consider the function $h_K:\mathbb{R}^d\rightarrow ...
2
votes
1answer
66 views

Solve given equation $4^{(x-2)(x+3)} - 64^{(x-3)} = 0?$

Solve given equation $4^{(x-2)(x+3)} - 64^{(x-3)} = 0?$ My attempt: I've attempted to solve this question, but isn't it impossible to solve, i.e has already been simplified completely? ...
0
votes
3answers
34 views

Obtaining expressions for functions

I'm new here, my first post :) And I am having issues with these questions. I have explained at the bottom what I have done so far, if you would be able to help me showing the working that'd be ...
1
vote
1answer
39 views

a set $X$ is infinite iff there isn't a bijection from $I_n\subset N$ to it

Consider the set: $$I_n = \{p\in \mathbb{N}; 1<p\le n\}$$ My book says that a set is finite when it's not empty or when there exists, for some $n\in \mathbb{N}$, a bijection: $$\phi: I_n\to X$$ ...
0
votes
1answer
27 views

find an injective function from a finite set to an infinite one, and a surjective inverse

I have to prove that there exists an injective function from $X$ to $Y$, being $X$ a finite set, and $Y$ an infinite set. I must also prove that there exists a surjective function from $Y$ to $X$. My ...
-4
votes
1answer
33 views

Let f be the function from $A = \{a, b, c, d\}$ to $B = \{e, f, g, h\}$ given by $f = \{(a,e), (b,f), (c,g), (d,g)\}$. [closed]

Let $f$ be the function from $A = \{a, b, c, d\}$ to $B = \{e, f, g, h\}$ given by $f = \{(a,e), (b,f), (c,g), (d,g)\}$. If $D = \{a,b\}$, what is $f(D)$? If $G = \{f,g\}$, what is $f^{-1}(G)$? If ...
-1
votes
1answer
52 views

Solve $f(-2x+3)=-f(4x-9)$ [closed]

Consider a function $f:\mathbb{R}\to\mathbb{R}$ that is bijective, has $f(\mathbb{R})=\mathbb{R}$ and $f(-1)=0$. Can the equation $f(-2x+3)=-f(4x-9)$ be fully solved? If yes, provide a proof. If not, ...
2
votes
1answer
24 views

Expectation of the fraction a random function covers its range

Preamble: The number of onto functions from a set of $m$ elements to a set of $n$ elements is, as stated in this answer, computed as follows: $$n!{m\brace n}\;.$$ Now, let's count the number of ...
3
votes
0answers
22 views

Let $A$ be a set with $m$ elements and let $B$ be a set with $n$ elements where $m,n\in \omega$ and $m>n$. If $f:A\to B$, then $f$ is not injective

So I am still learning how to work with infinite sets, and this particular problem is giving me some issues. Right now, I am trying to pick some $x_1,x_2\in A$ such that $f(x_1) = f(x_2)$ to serve as ...
1
vote
1answer
36 views

Finding a Transformation for a Sum of Exponentials

I am looking to see if it is possible to find a transformation $T_i(f(x))$ such that $$T_1\left(e^x+e^{ix}+e^{-x}+e^{-ix}\right)=e^x-ie^{ix}-e^{-x}+ie^{-ix}$$ ...
-1
votes
0answers
28 views

Function “all part” of $x$? [closed]

There is a simple math task that I need to solve: $D(f)=[0,1]$ Solve the $f([x])$. The problem is that I don't know what exacly means $[x]$ (all part of $x$)?
6
votes
3answers
154 views

$f>0$ on real line ; $f(x+y)\le f(x)f(y) , \forall x,y \in \mathbb R$ ; $f([0,1])$ is bounded set ; does $\lim_{x \to \infty}(f(x))^{1/x}$ exist?

Let $f: \mathbb R \to (0,\infty)$ be a function such that $f(x+y)\le f(x)f(y) , \forall x,y \in \mathbb R$ and $f$ is bounded on $[0,1]$ ; then does the limit $\lim_{x \to \infty}(f(x))^{1/x}$ exists ...
0
votes
1answer
23 views

Showing surjection

Suppose $\psi$ is a bijection $\psi: G\rightarrow H$ $\left ( g \right )\psi \mapsto h$ I want to show that $\psi^{-1}$ is also a bijection. $\psi^{-1}: H\rightarrow G$ $\left ( h \right ...
0
votes
0answers
20 views

inflexion points of a composition of functions

Let $f,g$ be two, smooth real positive and bounded functions over $\mathbb{R}^{+}$, with $f$ monotonously increasing and $g$ monotonously decreasing and both $f$ and $g$ have a single inflexion point. ...
-4
votes
1answer
127 views

Sorting the digits of $\pi$ [closed]

Given a function that sorts the digits of a real number, $\operatorname{sd}(r) \rightarrow r$, examples: $\operatorname{sd}(1.332) \rightarrow 1.223$ $\operatorname{sd}(32140) \rightarrow 1.234$ ...
-2
votes
0answers
10 views

A question on limits and integrals of continuous functions.

Assuming a function is continuous is it possible to do this, $$ \lim_{n\to\infty} \int_0^1 f\left(\frac{x_1+x_2+..+x_n}{n}\right) dx_1dx_2...dx_n = \int_0^1 ...
0
votes
2answers
39 views

Is this function one-to-one or onto?

Let $c:P(\mathbb{N})\to\mathbb{Z}$ be the function defined by $c(X)=−1$ if $X$ is empty and $c(X)=\min_{x\in X} x$ if $X$ is not empty. Is $c(X)$ one to one? Explain. I have asked this question ...
0
votes
1answer
32 views

Let f(x) = 2x. What is i. $f(\mathbb{Z})$? ii. $f(\mathbb{N})$? iii. $f(\mathbb{R})$?

I am struggling with this question and really need hints I wish if there is an upload file button here so that I can upload my solution to the question for check, since i faced difficulties typing ...
0
votes
1answer
24 views

I need someone to show me how to solve this input/output problem

Alright, so I have: $4y^3 = x$ And now I have to solve for $y$, where I can later use that equation to answer other questions I have. Can someone hint me out on how to solve for $y$ given the above ...
-1
votes
2answers
26 views

Let $A =\Bbb N − \{n^2 : n \in\Bbb N\}$. Construct a bijection from $A$ to $\Bbb N$.

To solve this function, I was thinking to use an index if that's possible: $f(x) = (i = 1$ if $x = 2$ --------$(i +1$ if $x_n = x_{n-1} + 2$ --------$(xi/x$ I see no other way to keep increasing the ...
1
vote
1answer
27 views

Let $f(x) = ax + b$ and $g(x) = cx + d$, where $a, b, c, d$ are constants. Determine for which constants $a, b, c, d$ it is true that $f ◦ g = g ◦$

I'm working on this question and this what I did I get $f•g(x) = f(cx+d)=a(cx+d) + b = acx +ad + b $ $g•f(x) = g(ax+b) = c(ax+b) + d = acx + cb + d $ So how to I get $f•g = g•f$?
0
votes
2answers
57 views

How to compute $\int \sqrt{x}\sin{\sqrt{x}}dx$?

How to compute $\int \sqrt{x}\sin{\sqrt{x}}dx$? I tried let $u=\sqrt{x}$ and $du=\frac{dx}{2\sqrt{x}}$ and apparently it doesn't work. Some better ideas?
0
votes
3answers
44 views

How to use mean value theorem to prove the inequality $|\sin{x}-\sin{y}|\le|x-y|$ for all $x,y\in\Bbb{R}$?

How to use mean value theorem to prove the inequality $|\sin{x}-\sin{y}|\le|x-y|$ for all $x,y\in\Bbb{R}$? So let us set $f(x)=\sin{x}$ then it's differentiable on $(x,y)$ and continuous on $[x,y]$. ...