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

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1
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0answers
25 views

Existence and uniqueness of a function generalizing a finite sum of powers of logarithms

I hope to find a proof of the following conjecture: $(1)$ For every $a>0$ there is a convex analytic function $f_a:\mathbb R^+\to\mathbb R$ such that: $f(1)=0$ and $\forall x>1,\ ...
2
votes
2answers
45 views

Prove that the function: f: $\mathbb{N} \mapsto \mathbb{Z}$ defined as f(n)= $\frac{(-1)^n(2n-1)+1}{4}$ is bijective.

Prove that the function: f: $\mathbb{N} \mapsto \mathbb{Z}$ defined as f(n)= $\frac{(-1)^n(2n-1)+1}{4}$ is bijective. This is rough. I've been staring at this one for a while now. I get stuck on the ...
-1
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0answers
42 views

List of simple, common functions with an incomplete domain or range on $\mathbb{C}$

This may seem like a strange question, but it's an interest of mine and I would appreciate the help of the community in addition to brainstorming on my own. As the question states, I'm looking for ...
-2
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0answers
32 views

algebra question MATH [on hold]

Find the indicated function and write its domain in interval notation. m(x) = , n(x) = x + 3, (m n)(x) = ? A) (m n)(x) = ; domain: [-5, ∞) B) (m n)(x) = (x + 3); domain: [-2, ∞) C) (m n)(x) = ...
0
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1answer
42 views

Find conditions on $m$ and $n$ that ensure that $f$ is a bijection.

Given: $X=\{0,1,2,...,m-1\}$, $f:X\to X$, $f(x)=nx \pmod m$. Find conditions on $m$ and $n$ that ensure that $f$ is a bijection. Progress It seems that $(m,n)=1$ but I can't prove that. I tried ...
3
votes
1answer
36 views

How many distinct values of floor(N/i) exists for i=1 to N.

Say we have a function $F(i)=\text{floor}(N/i)$. Then how many distinct values of $F(i)$ will exist for all $0 \leq i \leq N$ e.g. We have $N=25$ then. $F(1)=25$ $F(2)=12$ $F(3)=8$ $F(4)=6$ ...
2
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2answers
83 views

Is the function $y = a^x + b$ exponential?

What exactly is an exponential function? Some of the sources at which I looked said that it's a function where the rate of change at $x$ $(f'(x))$ is proportional to the value at that point $(f(x))$, ...
0
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0answers
39 views

Let $f:\Bbb{R}\to\Bbb{R}$ be a continuous function. Which of the following sets cannot be the image of $(0,1]$ under $f$ [duplicate]

Let $f:\Bbb{R}\to\Bbb{R}$ be a continuous function. Which of the following sets cannot be the image of $(0,1]$ under $f$? A. $\{0\}$ B. $(0,1)$ C. $[0,1)$ D. $[0,1]$ I think ...
0
votes
1answer
115 views

How many $f(x)$ are possible satisfying $f(x)=f'(x)$ and $f(0)=f(1)=0$.

Let $f:[0,1]\to\Bbb{R}$ be a fixed continuous function such that $f$ is differentiable on $(0,1)$ and $f(0)=f(1)=0$. Then the equation $f(x)=f'(x)$ admits how many solutions? The only solution ...
2
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0answers
48 views

How to show that $f$ can only have at most one root in $(a,b)$ with these conditions?

Let $f: [a,b]\rightarrow\mathbb{R}$ be a differentiable function on $(a,b)$. Suppose $f$ has the following property: If for an $x \in (a,b)$, $f(x)=0$, then $f'(x)>0$. The excercise is to show, ...
0
votes
1answer
28 views

A question about sets of limit points of continuous functions.

Let $f:\Bbb{R}\to\Bbb{R}$ be a continuous function and $A\subset\Bbb{R}$ be defined by $A=\{y\in\Bbb{R}:y=\lim\limits_{n\to\infty}f(x_n)$, for some sequence $x_n\to+\infty\}$. Then $A$ is ...
3
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0answers
40 views

How many such functions are possible?

Let $f$ be a function from $\{1,2,3,\dots,10\}$ to $\Bbb{R}$ such that ...
-3
votes
0answers
33 views

Let $A = \{1,2,3,4\}$, and $B=\{x,y,z\}$. [on hold]

a) Give an example of a function $f: A \to B$ that is onto b) Give a function $g: B \to A$ so that $f\circ g=iB$. c) Is your function $g$ one-to-one? Justify. Any help on this homework problem I ...
0
votes
2answers
21 views

Proving that the relation from the null set to the null set is a function

How would one prove that a relation that maps the null set to the null set is a function? I tried showing that the domain of the relation is the null set, but I'm unsure of where to proceed from ...
1
vote
2answers
36 views

Order of $f(n) = 4n + 6n^3 - 8n^5$

If a function $$f(n) = 4n + 6n^3 - 8n^5$$ then the order of $f$ is: The answer I have is $\log(n)$, but I'm not sure if it's right.
1
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4answers
28 views

Find the limit of function using Taylor series

Good evening, I'm somehow stuck on solving some easy exercises : $$\lim_{x\to\infty} x^{3/2}\bigl(\sqrt{x+1}+\sqrt{x-1}-2\,\sqrt{x}\bigr)$$
0
votes
1answer
35 views

$C^l$ diffeomorphism between a smooth manifold and a $C^k$ manifold

Let $M$ and $N$ be two Riemannian manifolds. $M$ is smooth while $N$ is $C^k$ manifold. Suppose there is a $C^l$ diffeomorphism between the two manifolds for $l \leq k$. Is it true that $N$ is also ...
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0answers
53 views

Bijection from $\mathbb{Z}$ to $\mathbb{Q}$ [on hold]

Can you explicitly tell me a bijection from $\mathbb{Z}$ to $\mathbb{Q}$. I need an explicit one. Thanks in advance.
0
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0answers
24 views

Analyze the variation of $f(x)=(1+\frac{1}{x})^{-K}$ $_2F_1((K-1)a,K,Ka,\frac{1}{1+x})$ w.r.t. $x$

Is there any way to analyze the variation(w.r.t. $x$) of the following function: $f(x)=(1+\frac{1}{x})^{-K}$$ _2F_1((K-1)a,K,Ka,\frac{1}{1+x})$, where $ _2F_1$ is the Gauss' Hypergeometric Function, ...
0
votes
1answer
14 views

Are the roots of a smooth function, a smooth function?

Let $f(x,y)$ be a smooth function. It is given that for every $x$ there exists at least one $y$ such that $f(x,y)=0$. Is this possible to select one such $y$ for every $x$, such that the $y$'s are a ...
3
votes
1answer
60 views

If $f(x)+2f(1/x)=3x$, find all $y$ such that $f(y)=f(-y)$.

The function $f(x)$ is not defined when $x=0$. This function has the property that $f(x) + 2f\left(\frac 1x\right) = 3x$. Find all such values of $y$ such that $f(y) = f(-y)$. (This means it is an ...
0
votes
2answers
53 views

$y=e^{-x}$ and $y=x$ point of intersection

How can I find the point of intersection of $y=e^{-x}$ and $y=x$ ? Here's the graph
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1answer
14 views

Value of $K$ so that $f(x)=3x^3+6x^2+KX-4$ has the same remainder when it is divided by $x-1$ and $x+2$ [closed]

Value of $K$ so that $f(x)=3x^3+6x^2+KX-4$ has the same remainder when it is divided by $x-1$ and $x+2$. Thanks for the help!
0
votes
1answer
20 views

How can I tell if the function $f(n)=2n$ on $\mathbb Z$ is one-to-one, onto, or both?

The domain of the function is the set of all integers. The codomain of each function is also the set of all integers. $$f(n) = 2n $$ I was thinking that the function is one-to-one but I don't know ...
1
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1answer
25 views

How to find $\lim\limits_{z \to z_0} \frac{{\overline z}^2-{\overline z_0}^2}{z-z_0}$

Fairly simple question, I want to find this limit $$\lim_{z \to z_0} \frac{{\overline z}^2-{\overline {z_0}}^2}{z-z_0}$$ The original question was to find the region at which the function ...
0
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0answers
13 views

Is this function upper hemi-continuous?

I want to determine whether a function $f(x)$ always attains a maximum value for some $x\in X\subset\mathbb{R}$, where $X$ is a compact set and $x\ge 0$. Thus, I need to check whether $f(x)$ is upper ...
0
votes
1answer
21 views

Relaxing Monotonicity of a Function $f:\mathbb{Z}\rightarrow \mathbb{R}$

Suppose a function $f:\mathbb{Z_+}\rightarrow \mathbb{R}$ fails monotonicity, but not by much. For example $f(2)= .3$ and $f(z)=1/z$ otherwise. Here there exists a single point where the function is ...
0
votes
2answers
27 views

how to find a spline function from given control points

consider having an n-amount of control points in 2D space, what's the best way to find the function passing through the start and end points while approximating the path according to the other given ...
0
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1answer
10 views

how to prove that a given function is univalent

I have to prove that following function is univalent $f(z) = z^2 +3z +1, ~|z|<1$ in complex plane. What I tried is: Let $f(z_1) = f(z_2)$ $\Rightarrow$ ${z_1}^2 +3z_1 +1= {z_2}^2 +3z_2 +1$ ...
2
votes
0answers
17 views

Proving a basic result about Holder continuous functions

Let $V$ be a open convex set. We will say that a function $m$ has the order of smoothness $p$ on $V$ with $p=l+\gamma$, where $l \geq0$ is an integer and $0<\gamma\leq1$ and will write $m\in ...
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2answers
48 views

Bijection between $\mathbb R^2$ and $(0,1)$ [closed]

I want to prove the sets have same cardinality: $\mathbb R^2$ and $(0,1)$ Please Help.
4
votes
1answer
75 views

Suppose that $f: A \to B$ and $g: B \to C$ are functions.

Suppose that $f: A \to B$ and $g: B \to C$ are functions. Prove the following: (a) If $g \circ f$ is injective, then $f$ is injective. Proof. Assume that $f$ is not injective. Then ...
1
vote
1answer
54 views

Using continuity to prove f is a constant function

Recently missed this problem on an exam. Just went to office hours to clarify what the proper proof was and wanted to see if, in attempting to repeat the problem, I can figure out if there are better ...
1
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1answer
19 views

Getting a function that passes through 'n' points

Is there a way to get a periodic function that passes through n arbitrary points?
0
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1answer
54 views

Find $\int_2^{2.2}f(x)\,\mathrm dx$ given $f(x)=x^4-3x^3+9x^2+22x+6$.

$f(x)=x^4-3x^3+9x^2+22x+6$. Find $\int_2^{2.2}f(x)dx$ by finding $f(x-2)$ This is in a non-calculator paper which is why $f(x-2)$ is meant to be obtained (it's supposed to made the maths possible to ...
0
votes
1answer
26 views

XOR function over binary vectors

I didn't really know how to name this question, it has been bothering me for some time: You are given n binary vectors of dimension $d: x_1,\cdots,x_n$; $x_i = x_{i_1},\cdots,x_{i_d}$. You are also ...
3
votes
2answers
56 views

If a function is Riemann integrable, then it is Lebesgue integrable and 2 integrals are the same?

Is is true that if a function is Riemann integrable, then it is Lebesgue integrable with the same value? If it's true, how to prove it? If it's false, what is a counterexample?
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0answers
27 views

Computationally Plotting an Equation

I have the following equation: $y^2 - log(y)^2 = 4log(x) + \frac{4}{x} + c$ and I want to plot this, real terms only, in the eqi-axis interval 0 to 5, for different values of c. Is there a general ...
0
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0answers
8 views

Normalization for argument of maximum function

is it possible to normalize the maximum function of a certain argument ? Means: Is that $\theta_{ML} = arg \max\limits_{\theta} \{ \sum \limits_{n=1}^{N} |w_n w^*_{n+N}| - \Big( \frac{SINR + ...
1
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1answer
49 views

On consequences of $\int_{0}^1f(x)x^ndx=0 , \forall n \in \mathbb Z^+\cup\{0\}$

If $f : [0,1] \to \mathbb R$ is a continuous function and $\int_{0}^1f(x)x^ndx=0 , \forall n \in \mathbb Z^+\cup\{0\}$ then is it true that i) $\int_{0}^1(f(x))^2dx=0$ ? ii) ...
1
vote
1answer
19 views

Taylor expansion of Airy function

We know that Taylor expansion is : $ f(x_0 + h) = f(x_0) + h f'(x_0) + .. \ $ I wish to expand the Airy function about it's first root , i.e , $Ai (c_1 - \epsilon ) = Ai (c_1) - \epsilon A_i'(c_1) ...
0
votes
2answers
41 views

How do you prove that a function is an injective/surjective function?

Basically I know that $$(f◦g)(x)=x$$ (from R to R for any x in R) How do I show that $f$ or $g$ are injective? and how do I show that they are surjective?
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1answer
28 views

Solution to functional equation and Cauchy

I want to show that the only continuos solutions to the functional equation $g(x)+g(y)=g(\sqrt{x^2+y^2})$ is $g(x)=cx^2$ where c is a constant i.e. c=g(1). I think that I maybe could rewrite the ...
1
vote
1answer
25 views

Proving if $\frac{3x+1}{x-1}$ is onto?

So, I have this function: $f(x)=\frac{3x+1}{x-1}$. So, in proving if it is onto, then by definition, for every b in B, there exists an a in A such that $f(a)=b$. So, let's solve or a. We get: ...
0
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1answer
27 views

Lower bound and upper bound functions

I am studying for a test and really need to know some examples of function with upper bound and lower bounds. I hope someone can be kind enough to help.Thank you Please give examples of function ...
3
votes
2answers
42 views

How to identify this function? $y = \log_2(y^{-1} + 4y)$

$$y = \log_2(y^{-1} + 4y)$$ How can I deal with the $y^{-1}$ and $4y$, also does identify mean find the domain, range and symmetry?
0
votes
0answers
29 views

How to prove a congruent to b (mod n) is a bijection?

I can prove it's an equivalence relation, but NO idea how to prove it's a bijection. I know I need to prove it's surjective/injective, but how do I establish it to even be a function?
1
vote
3answers
36 views

How to graph $g(x)=4^x-1$ and find its domain and range? [closed]

How to graph $$g(x)=4^x-1$$ and give its domain and range using interval notation? I have no idea what to do.
7
votes
3answers
361 views

What's the difference between a bijection and an isomorphism? [duplicate]

I don't understand what the difference is between a bijection and an isomorphism. They seem to both just be a invertible mapping. Is the set of all bijections a subset of isomorphisms? Or vice ...
1
vote
3answers
43 views

Smooth function to model a well or pit

In order to solve a Physics problem, I have to model an axisymmetric well or pit on a plane. I am looking for a function $f(x)$ that is smooth and monotonic in the domain $(0,1)$, where the origin ...