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

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21 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.
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1answer
24 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 ...
1
vote
5answers
63 views

Proving a function is surjective.

Let $X = \mathbb{R}$ and $Y = \{x ∈ R|x ≥ 1\}$. 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 function to equal y $G: ...
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vote
2answers
24 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: ...
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1answer
26 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?
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1answer
36 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?
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1answer
15 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?
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5answers
53 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
14 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 ...
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0answers
21 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 ...
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1answer
20 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
49 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 ...
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0answers
17 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 ...
3
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0answers
12 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 $Lip (\alpha, L_2(0,1))$ for $0<\alpha < 1$. $Lip(\alpha, L_2(0,1))$ is defined as the set of all functions ...
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1answer
13 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 ...
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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
39 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: $\int_0^1 p(x)\, dx = 1$ ; ...
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1answer
31 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 ...
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0answers
22 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\\ ...
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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 ...
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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 ?
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3answers
22 views

Limit of the function: x if x is rational and -x if x is irrational

The question is given as follows: Let $$g(x) = \begin{cases} x & x\text{ rational} \\ -x & x\text{ irrational}\end{cases}.$$ Prove that $\lim_{x\to 0}g(x) = 0.$ My first thought is to use ...
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1answer
35 views

Prove a functions is injective

Prove the function $f:\mathbb{N} \to\mathbb{N}$defined by $f(x)=2^x$ for all $x$ in $\mathbb{N}$ is one to one. Is my proof correct and if not what errors are there. For all $x_1,x_2$ $\in$$N$, ...
3
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2answers
30 views

what is the basic difference between a mapping and a function?

what is the basic difference between a mapping and a function? many say they are same but the opposite views are also seen. is mapping a restricted version of a function?
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1answer
17 views

Restricted Domain and Range

If i have $f|_A(x)=\psi(x)$ where I have $f:X\to Y$, and $\psi:A\to Y$ and i set it so i have $f(x)=x$, and i set that $X=Y=\{1,2,3,4,5\}$ and $A=\{1,2,3\}$ then for this i get that ...
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3answers
61 views

Suppose that $f ' (x)$ exists and $f(x)$ has two roots $x_1$ and $x_2$. Try to prove that:

Suppose that $f'(x)$ exists and $f(x)$ has two roots $x_1$ and $x_2$. Try to prove that: there is $\xi \in (x_1,x_2)$ such that $f(\xi)+f'(\xi)=0$. We cannot use the knowedge of integration.
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1answer
21 views

Sign of a function on interval [0,1]

Let $f(x)=(nx)^{-1+\frac{1}{s}}$ , $g(x)=(nx)^{-\frac{1}{2}}$ with $ 0\leq x \leq 1$ and $n\geq1$ Study the sign of $f-g$
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0answers
24 views

The set composed of domain and codomain of integrable function measure zero

There is this problem which I have constructed a plan to prove, and I am stuck. If anyone could see my plan and tell what is wrong about it I would be very thankful. Let $f: Q \to [0,1]$ be ...
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1answer
21 views

Algorithmic question regarding permutations

An algorithm i'm reading has the first step saying. The algorithm is a path enumeration algorithm which puts all permutations into lexicographic order. The algorithm works as follows Input: A ...
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2answers
18 views

Find the image of $A=(-2,1) \times [-2,2)$ under the function $f(x,y)=x^2y$

I have function $f(x,y)=x^2y$ and I have to find image $f[A]$ where $A=(-2,1) \times [-2,2)$ we have that $-2 < x < 1$ and $-2\le y<2$ $0 \le x^2 < 4 $ I claim that the image of ...
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vote
1answer
25 views

What is this function?

In python I made this function: def f(x): eqStr = '' for y in range(int(x)): eqStr += 'x**%s + ' % (y) eqStr += '0' return eval(eqStr) ...
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0answers
9 views

Is there any function analogous to Heaviside step function for boolean input?

I want to have a mathematical model in an analogous way with step function (for integers). To be more specific, I want a function (in theory, not in any programming language) such as function ("i same ...
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1answer
25 views

Let f be a map from R to R.Show that f’(a) is the derivative of f at a…read the question [on hold]

Let $f$ be a function from $\Bbb R$ to $\Bbb R$. Show that $f’(a)$ is the derivative of $f$ at $a$ if and only if $$\lim_{h\to0}⁡\dfrac{f(a+h)-f(a)- f' (a)h}{|h|}=0.$$
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1answer
30 views

Fixed point of a mapping

How to prove that every continuous $f:S^1 \to S^1$ such that $deg(f)\neq 1$ has a fixed point? One hint is that if $f(x)\neq x$ for any $x\in S^1$ then $f$ is homotopic to the antipodal map $a$ but I ...
0
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0answers
35 views

Why does Power Set represented by $2^x$ takes only $0$ or $1$ as values for $x$

While I was studying about Function Spaces I've seen an example of Function Space from function space of Power Set that tells that it(power set) maps from $X$ to $\{0,1\}$. I couldn't get how that ...
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0answers
7 views

Space $C^2(\overline{U})$ for open set $U$

Let $U$ be a bounded open domain in $\mathbb{R}^n$. Does the space $C^2(\overline{U})$ (the bar over $U$ means closure) mean the set of twice-differentiable functions $u$ such that $u, u_t, u_{x_i}$ ...
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2answers
19 views

Define f : Z/3Z → Z/3Z by f ([a]) = [2a + 1].

For this problem, I have to prove the function is well-defined, is surjective, and is injective. For seeing it is well defined, I have this: Assume [a1] = [a2] in the set of equivalence classes Z/3Z. ...
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0answers
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Suppose f: A → B. Define a relation ∼f on A by a ∼f b if f(a) = f(b).

So, as stated above, here is my question: Suppose f: A → B. Define a relation ∼f on A by a ∼f b if f(a) = f(b). First, I have to prove that ~f is an equivalence relation on A. So I need to show that ...
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0answers
16 views

Bijective functions on a finite set

Suppose that A is a finite set and f : A → A and g : A → A are functions. I need to prove that g ◦ f is a bijection if and only if f and g are bijections. So, could I say: Assume g of f is a ...
0
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1answer
23 views

Define $f:\mathbb{Z}/4\mathbb{Z}\to \mathbb{Z}/4\mathbb{Z}$ by $f([a])=[3a+1]$.

Define $f:\mathbb{Z}/4\mathbb{Z}\to \mathbb{Z}/4\mathbb{Z}$ by $f([a])=[3a+1]$. Prove that $f$ is well-defined, surjective and injective I don't really have a problem with figuring out if it's ...
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2answers
34 views

Finding a counterexample to a function proof

This is my proof: If f and g are surjective, then g ◦ f is surjective, with f: A $\to$ B and g: B $\to$ C. I have successfully proved this, but now I have to disprove the converse by finding a ...
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1answer
18 views

max and minimum qudratic function problem

A piece of wire $20$ metres long is cut into $2$ pieces and each piece is bent to form a square. Determine the length of the two pieces so that the sum of the areas of the two squares is a minimum. ...
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1answer
20 views

Problems Proving Injectivity and Surjectivity

I have these two functions, in which I have to prove or disprove they are injective and surjective: $f:[0,\infty) \to (0,\infty)$ by $f(x) = \frac{1}{x+1}$. $h:\mathrm R \to \mathrm R$ by $h(x,y) = ...
4
votes
1answer
49 views

Why do we have trigonometric functions besides $\sin(x)$?

Probably a terrible question, but I've been curious and can't come up with a reason besides convenience for myself with my limited knowledge. Why do we have $\cos(x)$, $\tan(x)$, etc. when all of ...
4
votes
2answers
50 views

Is $g : \mathbb R →\mathbb R$, $g(x) = |x|$ one-to-one and onto?

So, here is my function, in which I am to prove or disprove both if it is onto and one-to-one: Define $g : \mathbb R →\mathbb R$ by $g(x) = |x|$. For onto, can I say that it is not, because if we ...
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0answers
39 views

Does there exist a differentiable function $f:\mathbb R \to \mathbb R$ such that $f'$ is no-where differentiable on $\mathbb R$?

Does there exist a real valued differentiable function $f:\mathbb R \to \mathbb R$ such that $f'$ , the derivative of $f$ , is no-where differentiable on $\mathbb R$ ?
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0answers
27 views

Define $f : Z/4Z → Z/4Z$ by $f ([a]) = [3a + 1]$.

Define $f : Z/4Z → Z/4Z$ by $f([a]) = [3a + 1]$. (a) Prove that $f$ is a well-defined function. (b) Prove that $f$ is surjective. (c) Prove that $f$ is injective. I'm having trouble with this ...
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2answers
47 views

True or False. If f(-2)=2 then x+2 is a factor of f(x) [on hold]

True or False. If f(-2)=2 then x+2 is a factor of f(x). Explain your answer. Explain Using the remainder Theorem I believe the teacher said. For college algebra.
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1answer
32 views

What means $A \subsetneq X$ with A ~ X? [on hold]

How it is possible to have a subset A, which is $\neq$ to X and at the same time they have an equivalence relation ~? When $A \subset X$ therefore a $\in$ A is also a $\in$ X. With A ~ X ...
1
vote
2answers
33 views

Proof of uniform continuity of a rational function

Let $f\colon [0, \infty) \to \mathbb{R}$ be defined by $f(x)= \dfrac{4x}{1+x}$. Show that $f$ is uniformly continuous on $[0, \infty)$.