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

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5 views

Is it true that x(t)*\delta(t-nT) is nonzero iff n=t/T?

Note: T is just some constant and n is an integer. I'm trying to verify the steps.But I'm unsure that the statement is true since for n=5, the expression would be nonzero at 5T (assuming x(5) is ...
1
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0answers
12 views

Can we find a natural number $m$ such that $[A^{c^{t}}]=[A^{c^{m}}]$?

Let $t$ be a non natural namber. Can we find a natural number $m≠t$ such that $$[A^{c^{t}}]=[A^{c^{m}}]$$ where $[x]$ is the integer part of $x$ (the floor function)? Here $A>1$ nad $c>2$ are ...
-4
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0answers
43 views

Is this function is Invertible? [on hold]

Let f be a function from $\Bbb R\rightarrow\Bbb R$ with $f(x)=x^2 $ $f$ is not one-to-one and $f$ is onto function, is $f$ is invertible or is $f$ onto function?
1
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1answer
36 views

Calculus - Functions

How do I go about this question? Also how exactly will its graph be? $$ f(x) = 1 + 4x -x^2 $$ $$g(x) = \begin{cases} \max f(t) & x \le t \le (x+1) ;\quad 0 \le x < 3 \\ \min (x+3) & 3 ...
0
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1answer
46 views

Find consumer demand as a function of time, given the demand equation and price

An importer of Brazilian coffee estimates that local consumers will buy approximately $Q(p)= 4374/p^2$ kg of the coffee per week when the price is $p$ dollars per kg. It is estimated that $t$ weeks ...
6
votes
3answers
40 views

If $f(x) $ and g(x) are functions such that $f(x+y) =f(x)g(y) +g(x) f(y) $ then …

Question : If $f(x) $ and g(x) are functions such that $f(x+y) =f(x)g(y) +g(x) f(y) $ then $\begin{vmatrix} f(\alpha) & g(\alpha) & f(\alpha + \theta) \\ f(\beta) & g(\beta) & f(\beta ...
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0answers
7 views

Preimage of a natural morphism

Let $\mathbb R^n$ and $L$ be a additive subgroup of $\mathbb R^n$. COnsider the natural map: $p:\mathbb R^n\to\mathbb R^n/L$ If $X\subset \mathbb R^n$ then the preimage of $p(X)$ is $X$? Thank ...
3
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1answer
22 views

Function notation meaning: $f: \{a,b\} \to a$ - Zorich - MA I - p18

I have some notation I haven't seen before: $$f: \{a,b\} \to a\text{ and } g:\{a,b\}\to b$$ What does this mean? We are mapping from some $X=\{a,b\}$ to some $Y=a$? So pretty much we are always ...
6
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1answer
28 views

Is there always an equilibrium point in a field?

For instance, considering a set of planets represented as point masses that create a gravitational field, will there always, no matter what set of points, be a place where I can stand with no net ...
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2answers
38 views

Finding range for $\frac{1}{\exp(x^2)+3}$

What is the range of $$\large\frac{1}{e^{x^2}+3}$$ I know that the answer is $\dfrac{1}{4}\ge h(x)\gt0$, but how do I show it
2
votes
1answer
32 views

Verifing that $f$ is integrable

$f:[0,1]\times[0,1]\to \Bbb R$ be given by $$f(x,y) = \begin{cases} xy^2, & \ y\lt x^2 \\ x+2y, & \ y\ge x^2 \end{cases}$$ I need to show that $f$ is integrable. My idea is that to show ...
3
votes
0answers
43 views

Fubini's theorem application proof check

I have proven a problem but I am unsure whether it is correct because the proof seems so simple that I think I might be mistaken. Please be kind to comment on my proof and tell me whats wrong with it. ...
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votes
2answers
23 views

Find the probability density function of $Y=X^2$

Consider the random variable X with probability density function $$f(x)=3x^2$$ if $0<x<1$, and $$f(x)=0$$ otherwise. Find the probability density function of $Y=X^2$. This is the first question ...
0
votes
1answer
27 views

f & g don't have limit in x=a do f*g have limit? [on hold]

Hi can we example two functions that they don't have limit in x=a but f+g, f-g, fg & f/g have limit in x=a?
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1answer
9 views

Functions ( Natrual domain, Critical point and Differentiation)

If it would be possible for someone to run through these questions quickly it would be hugely appreciated. Im a bit rusty and having a small mind block. Thanks! Consider the function $f(x) = ...
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1answer
22 views

The relation of domain and image of a function and its inverse

Theorem: Let both $f$ and $f^{-1}$ be functions. $\newcommand{\dom}{\operatorname{dom}}\newcommand{\im}{\operatorname{im}}$ Then $\dom(f) = \im(f^{-1})$ and $\dom(f^{-1}) = \im(f)$. Let $f: X ...
0
votes
3answers
18 views

Functions that go up two and then down two

I'm trying to make a function that goes up two and then down two (kind of like sin(x) but without the curves). I keep drawing a blank on what I can do to even create this functions as I haven't done ...
-2
votes
2answers
15 views

Proving a Relation that is a Function by Division Algorithm [duplicate]

Let A=B=$\mathbb{N}$ R is: (a,b)$\in$R iff for some q$\in$Integers a=5q+b WHERE 0$\leq$b<5 Given a relation, show that it's a function. To Show: 1) $\forall$a$\in$A$\exists$b$\in$B((a,b)$\in$R) ...
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1answer
36 views

Is the relation on integers, defined by $(a,b)\in R\iff a=5q+b$, a function? [on hold]

Let $A=B=\mathbb N$. Relation $R$ is: $(a,b)\in R$ iff for some $q \in \mathbb Z$ we have $a=5q+b$ Given a relation, show that it's a function. To Show: $\forall a \in A \ \exists b \in B$ such ...
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votes
2answers
23 views

rational fractiona function solving for zero [on hold]

if you have a function $F(x)=\frac ab$ and you are asked to find the zero(s) of the function, why do you set the numerator equal to zero, and not the denominator?
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2answers
34 views

How do I find the zero(s) of a rational function?

I am doing homework and have been given this task: You have the function $$g(x)=\frac{2x^2-8}{x^2+4}$$ and I am asked to find the zeros of the function. My teacher shows the solution as $f(x)=0$ and ...
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1answer
26 views

Image, preimage and set operation in mappings

Not the best title but I don't know how to better describe it. So the image of a set is usually written as $f(B)$, my question is, can I use sets in the place of variables in the expression of my ...
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1answer
17 views

find image and inverse image of function

I have function $f:R\to R^2 , \ \ f(x)=<\cos 3x, \sin 3x>$ and I have to find image on the interval $(0, \pi]$ and inverse image $[0, +\infty) \times[0, +\infty)$ I think the image will be ...
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votes
3answers
12 views

find the Bijective function that answers the criteria: [0,1] -> [0,1) union [3,4]

find the Bijective function that takes elements of [0,1] (the numbers between 0 and 1 included) and matches exactly one element in the set [0,1) $\bigcup$ [3,4] (notice that 1 is not defined. the big ...
2
votes
2answers
69 views

If $f(2x-1) = 4x$, find $f(x)$

How can I solve these kind of problems? Can someone explain, please? $f(2x-1) = 4x$, find $f(x)$
2
votes
2answers
34 views

finding the period of $\sin(2x+3)$

I tried to find the period of $\sin(2x+3)$; looking for $p>0$ such that $ \sin(2(x+p)+3)=\sin(2x+3),$ for all $ x \in R$ which means: $ \sin(2(x+p)+3) - \sin(2x+3)= 0 ,$ for all $ x \in R$ ...
4
votes
3answers
66 views

Continuous functions satisfying $f(x)+f(2x)=0$?

I have to find all the continuous functions from $\mathbb{R}$ to $\mathbb{R}$ such that for all real $x$, $$f(x)+f(2x)=0$$ I have shown that $f(2x)=-f(x)=f(x/2)=-f(x/4)=\cdots$ etc. and I have also ...
0
votes
1answer
73 views

Questions about $f(n)=3+\frac{12}n$

Experimental Psychology: To study the rate at which animals learn, a psychology student performed an experiment in which a rat was sent repeatedly through a laboratory maze. Suppose the time in ...
2
votes
4answers
55 views

What does 'express in terms of $x$' mean?

For the following question : $f(x) = 2x^2 + 4x $ It asks me to express the following in terms of $x$: $f(-2x)$ What does the question mean by this? Does it mean make $x$ the subject?
-1
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0answers
25 views

Let $X = \mathbb{R}$ and $Y = \left \{ x \in \mathbb{R}\mid x ≥ 1 \right \}$. Define $G : X → Y$ by $G(x) = e^{x^2}$. Prove that $G$ is onto. [duplicate]

Let $X = \mathbb{R}$ and $Y = \left \{ x \in \mathbb{R}\mid x ≥ 1 \right \}$. Define $G : X → Y$ by $G(x) = e^{x^2}$. Prove that $G$ is onto.
1
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1answer
25 views

Determining if $f:\mathbb{R}\times\mathbb{R}\implies\mathbb{R}, f(x)=x-y $ is onto or surjective

Not sure how to determine this. $w\in\mathbb{R}, w=x-y\implies x=w+y$ and $y=x-w$. So that $f(x,y)=(y+w)-(x-w)=2w+y-x$. The thing is I think this is in Rng(f) but not sure how apparent that is or ...
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6answers
96 views

The function $G: x \mapsto 2^{x^2}$ maps $\mathbb{R}$ onto $\{ x \in \mathbb{R} : x \geq 1 \}$

Let $X = \mathbb{R}$ and $Y = \{x \in \mathbb{R} :x ≥ 1\}$, and define $G : X → Y$ by $$G(x) = e^{x^2}.$$ Prove that $G$ is onto. Is this going along the right path and if so how do get the ...
1
vote
2answers
29 views

How to solve reciprocal linear function

I was doing my homework and the last question was said to be a trickier one and if you could figure it out, then do it. It's bugging me as I want to understand how to do it. The question is as is: ...
0
votes
1answer
38 views

Consider $f(x) = \frac{2x^3-1+\sin x}{x^2-3}$. Show that $f (x) < 2x$ for most negative values of $x$.

Consider $$f(x) = \frac{2x^3-1+\sin x}{x^2-3}$$ Show that $f (x) < 2x$ for most negative values of $x$. How do I start this/ what concepts does this questions test?
0
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1answer
38 views

Find all real numbers $x$ such that: $\lfloor 7x\rfloor = 7$

I'm not quite sure how to approach this. Does $x$ have to be very small for it to work?
0
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1answer
16 views

Absolute value function homomorphism

$\theta:\mathbb R^*\to \mathbb R ^+ $ defined by $\theta(a)=|a|$ I know this function is an homomorphism but how do you prove it?
0
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5answers
56 views

Composition of two functions is not commutative

I have been always shown that the composition of two functions is, in general, not commutative with a counterexample. But can you give a more general proof of this statement (that is to say, one that ...
0
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0answers
17 views

How to determine the limit of a complex function

It is easy to show that a complex function doesn't have a limit as it approaches a certain point, but is there any way to know for sure whether any given complex function has a limit as it approaches ...
2
votes
1answer
27 views

Find a bijection, check if a given set is a function

I have problems with two exercises: $1)$ Find a bijection between $A$ and $B$. $$A=[0,1) \times[0,1)$$ $$B=\{{<x,y>}\in \mathbb R^2: x,y>0,\ x+y<1\}$$ $2)$ Decide if the given set is a ...
1
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1answer
25 views

Approximation of $f \in L^1_{loc}$

I am trying to prove the following statement: If $\Omega$ open in $\mathbb{R}^n$, $f \in L^1_{loc}(\Omega)$ (a set of all functions whose integrals on compact sets exist) and $\int_{\Omega}f\cdot g ...
3
votes
1answer
51 views

Hausdorff dimension of $\lim_{n\to\infty}\sin(2^nx)$

Calculate the Hausdorff dimension,$\dim_H$ of $$S=\{x\in(0,1):\lim_{n\to \infty}\sin2^nx=0\}$$ By definition We need to find the minimal $\alpha$ s.t $\sum_{i\in I}|U_i|^\alpha$ is minimal where ...
0
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0answers
18 views

Real analysis/cont-UC [on hold]

I have exam tomorrow and i need a help If $df/dx$ is bounded on any interval $E$ then $f$ is Uniformley continuous ? What about if $E$ is compact?Is it true or false ?Justify or give an example ...
4
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0answers
21 views

Function in Lipschitz space

I'm looking for a function that is in $W^{1,1}(0,1)$ but only in the Lipschitz space $\mathrm{Lip} (\alpha, L_2(0,1))$ for $0<\alpha < 1$. $\mathrm{Lip}(\alpha, L_2(0,1))$ is defined as the set ...
0
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1answer
20 views

Show uniqueness of a point

I use the banana function $F(x_1,x_2)=(1-x_1)^2+100(x_2-x_1^2)^2$ and I found the minimum point X to be (1,1). I need to show the uniqueness of that point. Could you please help me on how to show ...
0
votes
0answers
9 views

Systems of equations using taylor's series and find an upper bound

I have a function $g(a)=f_i(x+a(y-x))$ where a$\in$$\Re$ and x,y$\in \Re^d$. I consider $g'(a)=f'_i(x+a(y-x))(y-x)$ and $g''(a)=f''_i(x+a(y-x))(y-x)^2$ Then I have to plug them in the Taylor's series ...
2
votes
2answers
48 views

Maximum of parabolas at interval $[0,1]$

A family of parabolas $p(x)$ is given for $x \in [0,1]$ by coefficients $(a,b,c)$ , everything real-valued: $$ p(x) = a x^2 + b x + c $$ The area of the parabolas is normed: $\int_0^1 p(x)\, dx = 1$ ; ...
2
votes
1answer
35 views

How to prove that solution of ODE is even function?

Could you please give me some hint how to prove this statement: If $f(x)$ is solution of $y'=4x^3e^{-|y|}$ then $f(x)$ is even function. It is obvious that $f(x)$ increasing for all $x>0$ and ...
1
vote
0answers
28 views

finding a function whose values at certain points are defined

I have certain polynomials in a sequence from which I am trying to derive the common term. The polynomials are: $7k^2-8\\ 21k^4-368k^2-704\\ ...
1
vote
2answers
14 views

Given $f(x)= \frac1{4(x+4)^2}-2$ Find vertex, $ y$ intercept etc.

Given $f(x)= \frac{1}{4}(x+4)^2-2$ Find: vertex, $y$-intercept, $x$-intercepts (if any), axis of symmetry What I have so far: Vertex: $(-4,-2)$ $y$-intercept: $(0,2)$ $x$-intercept: $2$ Axis of ...
2
votes
0answers
37 views

Looking for differentiable function $f:\mathbb R \to \mathbb R$ whose derivative is nowhere continuous [duplicate]

Does there exist a differentiable function $f: \mathbb R \to \mathbb R$ such that its derivative $f'$ is nowhere continuous ?