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

learn more… | top users | synonyms (1)

1
vote
1answer
25 views

Don't know what this expanding periodic-ish function is

I plotted a function $c(x)$, which returns $3x + 1$ if $x$ is odd, and $x/2$ if $x$ is even. It's the Collatz conjecture. I get this interesting function. I don't know what it's called, so I can't ...
4
votes
3answers
76 views

Finding number of functions from a set to itself such that $f(f(x)) = x$

The questions states that $f: A\rightarrow A$ is a function which satisfies $f(f(x)) = x.$ We have to find the number of such functions with $A = \left\{1,2,3,4\right\}$. The given condition clearly ...
1
vote
1answer
39 views

Dude with taylor polynomial

Good night, i'm working with an problem of polynomial taylor, but i have a problem with the residue. Get a quadratic approximation $f\left(x,y\right)=\sin\left(x\right)\sin\left(y\right)$ near the ...
1
vote
2answers
37 views

$e^z=3z^5$ - Rouche's theorem

Question : Show that the equation $e^z=3z^5$ possesses five distinct real roots. In using the Rouche's theorem with the function $f(z)=-e^z+3z^5$ and $g(z)=-3z^5$, I succeeded to prove the ...
0
votes
0answers
8 views

Substituting into a function

$\hat{T}$ is a constant Are the parts I have underlined in green a mistake? Should they not be $C(\frac{p}{a}\hat{t}-\hat{T})$ as opposed to $C(\frac{p}{a}(\hat{t}-\hat{T}))$? If I have ...
1
vote
2answers
39 views

Preimage of sets, complement of sets, continuity of functions

I just got some simple questions in real analysis regarding preimage and complement of sets and continuity. Suppose $f:X\to Y$, then does $f^{-1} (Y\setminus F)=f^{-1} (Y)\setminus f^{-1} ...
-1
votes
2answers
46 views

Example of a function that converges to 0 pointwise but integral is 3/2?

Give an example of a sequence of continuous functions $(f_n)$, $f_n : [0, 1] \to \mathbb{R}$ that converges to zero pointwise, and such that the integral of each function within the given domain is ...
0
votes
3answers
23 views

How to determine intervals where $f$ is greater than $g$?

I have two functions, $f(x) = 2x$ and $g(x) = \frac{x^3}{3}$. I solved for $x$ where $f = g$, finding $x = \pm 6^{1/2}$, then solved for $x$ where $f > g$, $x > \pm 6^{1/2}$, and where $f < ...
-3
votes
2answers
55 views

Periodic functions $f(x)=\sin{x}+\cos{x}$ [on hold]

Find period of functions $$f(x)=\sin{x}+\cos{x}$$and $$f(x)=|\sin{x}|$$ I need some hints and directions.
0
votes
0answers
16 views

Lower semi continuity for the norm of the speed

Let $\Omega$ be a smooth bounded open domain in $\mathbb R^d$. Let $\gamma_n: [0,1]\to \overline{\Omega}$ be a sequence of Lipschitz functions which converges uniformly on $[0,1]$ to a Lipshitz ...
3
votes
3answers
140 views

Uses of step functions

My highschool teacher has informally told us about what continuity is and used step functions as an example of a discontinuous function. The Wikipedia page for it links to a lot of other kind of step ...
0
votes
2answers
37 views

What exactly does f'(x)=0 imply from the definition of differentiability?

Let f be a real valued function satisfying $|f (x) −f (a)| ≤ C|x−a|^γ$, for some γ > 0 and C >0. (a) If γ = 1, show that f is continuous at a; (b) If γ > 1, show that f is ...
5
votes
1answer
57 views

Would such a function be of any importance (primality test)?

While experimenting with some Maths, I came up with a really cool function. Let's call this function $\space \beta \space$. Which is a function of a real variable $\space r \space $. Here is the ...
0
votes
0answers
60 views

Very difficult functions to prove with O notation

I am trying to prove some O notations as is it one of the tasks for my assignment in my course in algorithms and data structures. First of all I'd like to be sure that I got the "recipe" right. I use ...
0
votes
1answer
25 views

Function of metric with a fixed point

I'm trying to prove that given a metric space $(X, d)$, for a fixed $x\in X$, define the function $g(y)=d(x,y)$, then $g(y)$ is continuous, using triangle inequality. My first question is that can I ...
0
votes
1answer
16 views

Isometries in $\mathbb{E}^2$

We define $\mathbb{E}^2 = (\mathbb{R}^2, d_E)$, where $d_E$ is the usual Euclidean metric on $\mathbb{R}^2$. We say that a function $f:X \rightarrow Y$, where $X, Y$ are metric spaces, is an isometry ...
12
votes
2answers
644 views

Solve these functional equations: $\int_0^1\!{f(x)^2\, \mathrm{dx}}= \int_0^1\!{f(x)^3\, \mathrm{dx}}= \int_0^1\!{f(x)^4\, \mathrm{dx}}$

Let $f : [0,1] \to \mathbb{R}$ be a continuous function such that $$\int_0^1\!{f(x)^2\, \mathrm{dx}}= \int_0^1\!{f(x)^3\, \mathrm{dx}}= \int_0^1\!{f(x)^4\, \mathrm{dx}}$$ Determine all such ...
0
votes
1answer
17 views

Composition of functions Discrete Math question

How do I do this? All help is appreciated! Would prefer a step by step tutorial but any help is ok :) Let $h= g\circ f\circ g$ where $f \colon \mathbb R \to \mathbb Z$ is the floor function and ...
1
vote
2answers
21 views

Changed codomain of inverse trigonometric functions

If codomain of $\arcsin(x)$ is $(\pi/2 , 3\pi/2)$ and codomain of $\arccos(x)$ is $(\pi , 2\pi)$ then what should be $\arcsin + \arccos$ equal to ? I thought of putting $x = \sin \theta$ But then ...
0
votes
0answers
23 views

Runtime of Algorithms (Recurrence&Induction)

Two algorithms are given: $$T_A(n) = (\log_4(n) + 1) \cdot n\quad\text{and}\quad T_B(n) = 4 T_B\left(\frac{n}{4}\right) + n^\alpha$$ $$T_B(1) = 1; \alpha \in \mathbb R_+; n = ...
2
votes
1answer
43 views

Functions: Relations

I'm stuck on this question and would appreciate the solution, thanks! Let $R$ and $S$ be the relations on $A = \{1,2,3,4,5\}$ given by$$R = ...
1
vote
0answers
30 views

Why can we calculate the Fourier series of $x^2$ in any interval $[-l,+l]$?

We know that a function must satisfy Dirichlet's Conditions before it can be expanded in Fourier series. And Dirichlet's Conditions strictly require a function to be periodic in the interval in which ...
1
vote
0answers
15 views

Cardinality of strict extrema of a real function

I recently encountered a problem seeking to prove that a real function can only have a maximum number of #$ \mathbb{N}$ strict maximums. It may be that I have copied the problem improperly since there ...
-2
votes
1answer
32 views

If f(x) is continuous on $[a,b],$ differentiable on $(a,b)$, and $f'(x)\neq 0\ \forall x\in(a,b),$ then f'(x) is stable. [on hold]

If $f(x)$ is a continuous function on $[a,b]$ and differentiable on $(a,b)$, and $f'(x)\neq 0\ \forall x\in(a,b)$ then $f'(x)$ is stable $\left(\text{i.e.},\ f'(x)<0\ \ \text{or}\ \ ...
2
votes
1answer
25 views

Invertible Uniform “PseudoRandom” Function

Perhaps this is better suited to a cryptography stack exchange, but I thought I'd try in mathematics in case this question is more obvious than I initially thought. I'm looking for a function ...
-1
votes
1answer
28 views

Integration/Functions Question [closed]

I'm having trouble with this question, especially the integration in parts (f) and (g). Could someone please solve and explain? Thanks!
1
vote
3answers
27 views

How to phrase a proof of a function from a set A to a set B

Here is a problem: Let $f \subseteq A \times B$ be a function. In many situations you may want to restrict the domain of $f$ or expand its range. If $C \subseteq A$ then define the restriction of $f$ ...
0
votes
0answers
17 views

How to represent the function of variables?

I have a function as $$E=\int_\Omega -\log\big( p_i(x)\big) dx$$ where $p_i(x)$ is density distribution which estimated by Parzen window method. $p_i(x)=\frac{1}{\Omega_i} ...
-1
votes
1answer
46 views

If $\lim_{x\rightarrow\infty}f(x)=0$, does $||f||_{L^2}=0$? [closed]

If $\lim_{x\rightarrow\infty}f(x)=0$, does $||f||_{L^2}=0$? Thank you very much.
0
votes
1answer
31 views

Can someone draw a plot for this function?

Can someone draw a plot for this function? $ f(x) = \begin{cases} \sum_{i=2}^{x}\left(\frac{\prod_{k=1}^{i-1}\left(2k-1\right)\,\cdot\,-\left(-\frac{1}{2}\right)^{i}}{i!}\right) + \frac{3}{2} & x ...
3
votes
1answer
19 views

Combination of even and odd functions

Can someone help me how to show that any function $f(x)$ defined on a symmetrically placed interval can be written as a sum of an even and a odd function? What is the special role played by ...
0
votes
1answer
26 views

Proof of Interceting Lines

I have this practice problem from a final exam study guide. Let $f,g$ be continuous on $[a,b]$ and $f(a)>g(a)$ but $g(b)>f(b)$. Prove that $\exists c \in [a,b]$ such that $f(c)=g(c)$. My ...
1
vote
1answer
27 views

Proving that $d(f,g)=\|f-g\| = \sup \limits_{0\leq x \leq 1} |f(x)-g(x)|$ is a metric on $X=C_b[0,1]$

Following Proving that $d(f,g)=\|f-g\| = \sup \limits_{0\leq x \leq 1} |f(x)-g(x)|$ is a metric on $X=C[0,1]$ I would like to prove that the same is true for bounded functions on $[0,1]$ ...
8
votes
1answer
58 views

Prove that $\exists x_0, x_1\in (0,1)$, such that $\frac{f'(x_0)}{x_0}+\frac{f'(x_1)}{x_1^2}=5$

Let $f:[0,1]\to\mathbb{R}$ be a differentiable function, such that $f(0)=0$ and $f(1)=1$. Prove that there exist different $x_0, x_1\in (0,1)$, such that ...
-1
votes
0answers
15 views

Functions and set theory question [duplicate]

$f:R\rightarrow R$ be a one to one function if and only if for any A,B⊂$R$ $$f(A\cap B)=f(A)\cap f(B)$$ I have to prove this and I don't have any kind of idea how to prove this. Please help me to ...
0
votes
0answers
22 views

Rate/Flow analysis using integration

I'm given a problem that a pump is removing oil at a rate modeled by the function $R(t)=2+cos(\pi t/11)$ over the interval t=[0,6]. And I'm supposed to find the volume of oil removed in the 6 hour ...
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
25 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?