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

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5 views

functions and relations,bijection

f:R-->R defined by f(x)= 5x^3+3.is it onto. according to me, if y=5x^3+3 ;x = cube root of (y-3)/5 then x = cube root of (y-3)/5 is not an element of 'R' for ...
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1answer
12 views

functions and the commutative property

with regard to vector spaces of functions. How do I know if the commutative property holds for a set of functions. especially if the vector space includes an infinite set. for instance, for the ...
1
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2answers
37 views

Why the continuity of $f$ is not a necessary condition?

I am quite new to functions and continuity, and now I am reading the slides regarding the intermediate value theorem, which is related to continuity of functions. While reading, I found the ...
2
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2answers
17 views

Increasing/Decreasing intervals of a parabola

I am being told to find the intervals on which the function is increasing or decreasing. It is a normal positive parabola with the vertex at $(3,0).$ The equation could be $y = (x-3)^2,$ but my ...
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1answer
38 views

Find $f(x)$ given $f, g$ such that $\,f(0) =2,\, g(0) =1, \, f'(x) = g(x),\, g'(x) = f(x)$.

Let $f$ and $g$ be functions satisfying: $$\begin{align} f(0) & =2\\ g(0) &=1 \\ f'(x) &= g(x) \\ g'(x) & = f(x) \end{align}$$ Find $f(x)$.
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0answers
9 views

Marginal distribution for two continous random variables

Let $f_{XY} (x,y) = a\;if\: x ∈ [-1, 0] \;and\; y ∈ [x^2, -x] $ or $f_{XY} (x,y) = 0 \;if\;x \notin [-1, 0] \;and\; y \notin [x^2, -x] $ is the distribution function for the random variables X and Y . ...
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1answer
19 views

What is $h^{-1}(L)$, for $L$ a regular language and $h$ a homomorphism?

Let $L = L((00 + 1)∗)$ and $h : \{a, b\}^* \to \{0, 1\}^*$ be defined by $h(a) = 01$ and $h(b) = 10$. What is $h^{−1}(L)$? In this context "$+$" means "$\cup$". So the language $L$ is all the ...
3
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1answer
54 views

Functions $f: \Bbb Q_{+} \to \Bbb Z$ such that $f(\frac{1}{x}) = f(x)$ and $(x+1)f(x-1)= xf(x)$

How to find all functions $f: \Bbb Q_{+} \to \Bbb Z$ such that $f(\frac{1}{x}) = f(x)$ and $(x+1)f(x-1)= xf(x)$ for $x> 1$?
2
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1answer
20 views

given the following two conditions, find $f(x,y)$

Suppose that a function $f$ defined on $\mathbb R^2$ satisfies the following conditions: $f(x+t,y)=f(x,y)+ty$; $f(x,t+y)=f(x,y)+tx$; $f(0,0)=k$; then for all $x,y \in\mathbb R$, $f(x,y)=$ a) ...
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1answer
29 views

domain of $\sqrt {\cos^{-1}(\cos x)-\lfloor x\rfloor} $

Here is my question where I got stucked. The domain of $\sqrt{\cos^{-1}(\cos x)-\lfloor x\rfloor} $ where $\lfloor \cdot\rfloor$ denotes the greatest integer function (floor function).
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0answers
7 views

Lagrange Multiplier, Boundary

In many cases when we have to optimize a function under a constraint, i.e $f(x,y)=e^{-xy}$ with constraint $x^2+4y^2 \le1$, Lagrange multipliers only help with finding the extreme values at the ...
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3answers
25 views

Functions - Inverses of graphs.

The question reads: sketch the graph of y=-3-x along with its inverse. From calculating the equation of the inverse graph, I come to y=-3-x, using the swap method. I then tried to plot both graphs ...
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1answer
19 views

Let $f\colon [a,b]\to\mathbb R$ is continuous and $G(x,t)=t(x-1)$ when $t\leq x$ and $x(t-1)$ when $t\geq x$.

Let $f\colon[a,b]\to \mathbb R$ is continuous and $$G(x,t)=\begin{cases}t(x-1)&\text{when $t\leq x$,}\\x(t-1)&\text{when $t\geq x$.}\end{cases}$$ Let $$g(x)=\int_0^1f(t)G(x,t)\,\mathrm dt.$$ ...
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2answers
33 views

What is the function $w(x) = 4 + \sqrt[3]{x}$? [on hold]

I need to know how to go about determining if this function is even, odd, or neither
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2answers
44 views

Differentiability and continuity at the origin of piecewise defined $g(x,y) = y-x^2$, $y+x^2$, or $0$

$$g(x,y)= \begin{cases} y-x^2, & y\ge x^2\\ y+x^2, & y\le -x^2\\ 0 & \text-x^2\le y\le x^2 \end{cases}$$ I need to find all the directional derivatives at the origin in the tangent ...
2
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3answers
77 views

Number of functions verifying $f(f(x))=f(x)$.

Find the number of functions $f:\{1,2,3,4\}\to \{1,2,3,4\}$ that verify $f(f(x))=f(x)$. I'm not sure if the answer is $41$ or $29$.
4
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6answers
70 views

$f:[a,b] \to R$ is continuous and $\int_a^b{f(x)g(x)dx}=0$ for every continuous function $g:[a,b]\to R$

$f:[a,b] \to R$ is continuous and $\int_a^b{f(x)g(x)dx}=0$ for every continuous function $g:[a,b]\to R$ with $g(a)=g(b)=0$. Must $f$ vanish identically? Using integration by parts I got the form: ...
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0answers
26 views

fn converges to f pointwise where all functions fn are bounded and f is unbounded: Is this example correct?

I am looking to find a sequence of functions $f_n$ that converges to a function $f$ pointwise, where all functions $f_n$ are bounded, but $f$ is unbounded. I have thought of an example where the ...
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1answer
10 views

What other types of distributivity are there?

When I say ‘Distributivity,’ I mean the way a number $x$ can be ‘Put in to’ some other function or the like. For example, to distribute $x$ into $\text{id}_y$, you simply have ...
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1answer
58 views

Solving the functional equation $f(x^2+f(y))=(f(x))^2+y$

Find all $ f : R\rightarrow R $ such that $f(x^2+f(y))=(f(x))^2+y, \forall\text{ x,y}\in R$ Thanks in advance!
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1answer
10 views

Writing a function in a part that is linearly dependent on the dependent variable

Let's say I have the function $f = f(x)$, Under what conditions am I able to write this function as follows: $f = c + h(x)x$ where $c$ is a constant, and $h(x)$ is some function depending on $x$.
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1answer
37 views

Evaluating $\lim_{n\to\infty}\int_0^1x^nf(x)\,dx$. [duplicate]

Let $f$ be a continuous function on [0,1]. Evaluate $$\lim_{n\to \infty} \int_0^1 x^nf(x)dx$$ My approach : Consider $\int x^nf(x)dx = \frac{f(x)x^{n+1}}{n+1} - \frac{1}{n+1}\int x^{n+1}f(x)dx$ ...
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1answer
58 views

Proof of Hyperbolic Functions

Find the proof:  (a) Use the definitions cosh(x)= 1/2(ex +e^−x) , sinh(x)= 1/2(e^x − e^−x) to express sinh(x + y) and cosh(x + y) in terms of cosh(x), sinh(x), cosh(y) and sinh(y). (b) Using the ...
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2answers
45 views

Can a real continuous bounded function on $ \Bbb{R}^{2} $ be expressed as a finite sum of products of real continuous functions on $ \Bbb{R} $?

Can a real-valued continuous bounded function on $ \Bbb{R}^{2} $ always be expressed as a finite sum of products of real-valued continuous functions on $ \Bbb{R} $?
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0answers
24 views

Squares and Square Roots [on hold]

what are the laws/principles/rules to determine when to square or take a square root of a variable? To say it another way, how do you determine when you need to square or take a square root of a ...
5
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2answers
174 views

Why is $\sinh$ often pronounced “shine”?

I talked to some guys from the UK and they told me that they would pronounce $\sinh$ as "shine". I am not a native english speaker so I don't know, but in my country we call this function "sintsh" ...
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1answer
14 views

Precision needed in definition of unboundedness

This is a quick question about what it means for a function to be unbounded. Does it mean that the function tends to + or - infinity, or does it just mean that it has no limit?
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1answer
19 views

How can this function be considered to have a saddle node bifurcation?

Say I have the function $f(x,\mu) = (1 + \mu)x − x^2 − 0.1$. By definition a Saddle Node bifurcation occurs if: $f_{\mu_0}(0) = 0$ $f'_{\mu_0}(0) = 1$ $f''_{\mu_0}(0) \neq 0$ $\frac{\delta ...
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1answer
33 views

How to generate integer random numbers that equal to another random number?

I am running a simulation in Excel, and need to generate a group of integer random numbers summing up to another random integer, how can I possibly do it? For instance I have an integer random number ...
2
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2answers
35 views

Can we solve recurrence relations indexed by R?

Recurrence relations are equations with functions from $\mathbb N\rightarrow \mathbb N$. For example given $a:\mathbb N\rightarrow \mathbb N$ (write $a(n)$ as $a_n$) solving the recurrence relation ...
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1answer
36 views

e Function Parity

I have look at the plot of the function $$(1+\frac{1}{x})^x$$ Is it an odd or even function? it seems like one flip on the Y-axis and on flip on the X-axis but unlike odd it is a flip upward Aren't ...
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1answer
31 views

Some conditions on $\tilde f(x,y)=\begin{cases}\displaystyle g(x,y) & \text{if }(x,y)\not=(0,0), \\ 0 & \text{if } (x,y)=(0,0).\end{cases}$

The following function $$f(x,y)=\begin{cases}\displaystyle\frac{x^2 y^2}{x^2+y^2} & \text{if }(x,y)\not=(0,0), \\ 0 & \text{if } (x,y)=(0,0).\end{cases}$$ is differentiable in the origin and ...
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1answer
23 views

Given that $f(x)=\frac{5}{x-8}$ and $g(x)=\frac{9}{x+9}$, find [on hold]

(a) $(f+g) (x)=$ (b) $(f–g) (x)=$ (c) $(fg) (x)=$ (d) $(f/g) (x)=$ I don't know how to calculate the result of (x) .
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1answer
59 views

If $f(U)=0$ then what is possible?

Let , $U=\left(0,\frac{1}{2}\right)\times \left(0,\frac{1}{2}\right)$ and $V=\left(-\frac{1}{2},0\right)\times \left(-\frac{1}{2},0\right)$ and $D$ be the open unit disk centered at origin of $\mathbb ...
0
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0answers
11 views

Shifting a series of functions whilst maintaining symmetry

I have a function y = a*Exp[-(x - b)^2/2*c^2] + d*Exp[-Abs[-e*x]] + f Which is symmetrical when the coefficient of b is equal to 0 however it loses symmetry as ...
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1answer
16 views

The Average Rate of Change of Function

Here is the question: The following chart shows the growth of a crowd at a rally over a 3 h period. Time (in hours): $ 0.0, 0.5, 1.0, 1.5, 2.0, 2.5, 3.0 $ Number of people: $0 , 176, 245, 388, 402, ...
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0answers
20 views

Eliminate the arbitrary Function of PDE

I need to solve this problem; Eliminate the arbitrary function $f$ from the equation: $f(x^2+y^2+z^2,z^2-2xy)=0$ I try this solution $u= x^2+y^2+z^2, \quad $ $\quad v=z^2-2xy, \quad$ so ...
2
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0answers
30 views

Inversion of a pairing function

I was reading this question on this site and I saw that the following pairing function was mentioned (a modified version of Cantor function): $$\langle x, y\rangle = x * y + ...
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1answer
23 views

Can I have a critique of this set theory proof/Advice on a similar proof?

This is an exercise from Mendelson's Introduction to Topology. The first part is to prove, given a function $\ f:A \rightarrow B$, that $\ X \subset f^{-1}(f(X))$ for all $\ X \subset A$. Here's my ...
3
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4answers
59 views

Prove that $[a/b]+[2a/b]+[3a/b]+…+[(b-1)a/b]=(a-1)(b-1)/2$

If a and b are positive integers with no common factor how to show that $[a/b]+[2a/b]+[3a/b]+...+[(b-1)a/b]=(a-1)(b-1)/2)$,where [.] denotes the greatest integer function? Im not able to understand ...
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1answer
17 views

Showing that a given function is convex.

I am trying to show that the function $f(x,\vec{y})=\alpha\ln(1+\exp(x+\vec{y}\cdot \vec{z}))+(1-\alpha)\ln(1+\exp(-x-\vec{y}\cdot\vec{z}))$ is a convex function of $(x,\vec{y})$ (where ...
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2answers
106 views

How do I prove that $ f(n) = (n + 1)! - 1 $ is an injective function?

I have this problem: Consider the function $f : \mathbb{N} \rightarrow \mathbb{N}$ defined, for every $n \in \mathbb{N}$, by $$f(n) = (n+1)! - 1$$ Prove that $f$ is injective. How do I ...
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1answer
39 views

Expanding function into Maclaurin series

How to expand that function below into Maclaurin series? Is it even possible? $$ f(x)=x^2+\ln\left(\frac{2x-3}{5-3x}\right) $$ I know that expanding into Maclaurin series requires function class ...
1
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1answer
21 views

Quadratic function with positive integral coefficients problem

Here is the problem statement: Let $f(x)$ is a quadaratic expression with positive integral coefficients such that for every $\alpha, \beta\; \epsilon\; \Re$, $\beta>\alpha$, $\int_\alpha^\beta ...
5
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0answers
40 views

Show that $1^k+2^k+…+n^k$ is $O (n^{k+1})$.

Let $k$ be a positive integer. Show that $1^k+2^k+...+n^k$ is $O (n^{k+1})$. So according to the definition of big-$O$ notation we have: $$1^k+2^k...+n^k ≤ n(n^k) = n^{k+1}$$ whenever $n>1$ Is ...
2
votes
0answers
44 views
+50

Find Functions That Can Be Inverted from Their Sums

I have the following situation:$$ f_1(x_1) + f_1(x_2) + f_1(x_3) + \cdots + f_1(x_n) = c_1\\ f_2(x_1) + f_2(x_2) + f_2(x_3) + \cdots + f_2(x_n) = c_2\\ \vdots\\ f_n(x_1) + f_n(x_2) + f_n(x_3) + ...
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2answers
23 views

shown that f is even

lets one function $f:\mathbb{R}-\{0\}\mapsto\mathbb{R}$ where $\mathbb{R}$ is the set of reals, lets $f$ such that $f\left(\frac{a}{b}\right)=f(a)-f(b)$ for every $a$ and $b$ in belonging to the ...
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1answer
29 views

A Problem Distribution Function

If I have a probability density function like this $w(x) = 1 - |x| $if $|x| \leq 1$ or $ w(x)=0$ if $|x|\geq 1$, what's the value of the distribution function F(x)? I mean that I calculated ...
0
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1answer
24 views

Manually plotting some particular graphs

How to plot graphs like these manually: 1) $f(x)=\ln(1+x^2)$ 2) $f(x)=\frac8{2+x^2}$ 3) $f(x)=\frac{\sin x}{\sqrt{1+\tan^{2}x}}+\frac{\cos x}{\sqrt{1+\cot^{2}x}}$ I have no idea how to plot the ...
1
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2answers
28 views

Continuity of $\mu(t)=\inf\{x \in \mathcal C : \kappa(x)=t\}$.

Let $\Delta = \{ 0, 1\}^{\mathbb N}$ be a Cantor set. Define $\theta : \Delta \to [0,1]$ by the formula $$\theta(x_1,x_2,\dots) = \sum_{n=1}^\infty \frac{2x_n}{3^n}.$$ Denote $\mathcal C = ...