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

learn more… | top users | synonyms (1)

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
29 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
14 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
31 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 [on hold]

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 $||f||_{L^2}=0$? [on hold]

If $\lim_{x\rightarrow\infty}f(x)=0$, does $||f||_{L^2}=0$? Thank you very much.
0
votes
1answer
30 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
26 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. [on hold]

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
43 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?
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. [on hold]

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
39 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
65 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
33 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 ...
-3
votes
0answers
16 views

Definition: f : A ! B is one-to-one if 8t 2 B, 9 at most one s 2 A such that f(s) = t. [closed]

The question ask me to prove by definition of one-to-one if this statement is one-to-one? I prefer False since it has nothing to do with one-to-one function. I'm I correct.