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

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0
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1answer
27 views

$|f(z)|\le\frac{M}{|z|^{\alpha}}$ for all $z\in U_r(0)\setminus \{0\}.$ Why is $0$ a removable singularity of $f$?

Let $0<r<1$, $f:U_r\setminus\{0\}\to\mathbb{C}$ holomorphic. Let $\alpha <1,\; M\ge 0$ such that $$|f(z)|\le\frac{M}{|z|^{\alpha}}$$for all $z\in U_r(0)\setminus \{0\}$. Prove that $0$ is a ...
0
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1answer
10 views

Writing the function to maximize volume or a cylinder

A rectangular piece of paper is curled into a cylinder with two open circles on each side. The perimeter of the piece of paper is 124 inches. What is a function that could be written to find the ...
2
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1answer
37 views

Multiple choice question about the continuity of sum and product of functions

Which of the following statements is/are not True below $A.$ if the sum $f(x)+g(x)$ is a continuous function at $x=a$, then both $f(x)$ and $g(x)$ are individually continuous at $x=a$ $B.$ if the ...
3
votes
2answers
59 views

Prove or give a counterexample for $f(x)$

Got this from my Real Analysis problem set: Suppose $f(x)$ continuous on some open interval $I$, and $c$ is maximum point for $f(x)$ inside this interval. Is it true that that $f(x)$ is ...
0
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2answers
26 views

Find the range of $f(x) = x^2$ defined on the domain $-1 \leq x \leq 4$

$f(x)=x^2$ for $-1 \leq x \leq 4$ So I put everything to the power of $2$ and got: $$1\leq x^2 \leq 16$$ However the answer should be: $$0 \leq f(x) \leq 16$$ How can I get that?
2
votes
4answers
49 views

Elementary symmetric polynomial task with three variables

Can anyone help me to wite this as sum or product of elementary symmetric polynomial. $$\frac xy+\frac yx +\frac xz + \frac zx +\frac yz + \frac zy =7$$ I tried to set under one fraction, but I ...
0
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0answers
7 views

A function basis for unimodal functions.

Fourier (trigonometric) basis is good for representing periodic functions. Polynomial basis is good for representing periodic functions near a particular point. Wavelet basis might be good for ...
1
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5answers
76 views

How to plot the graph of function $f(x) = \sqrt{8\sin^2x+4\cos^2x-8\sin{x}\cos{x}}$?

How to plot the graph of function $$f(x) = \sqrt{8\sin^2x+4\cos^2x-8\sin{x}\cos{x}}$$ Is it even possible ? When I tried it the function compressed into $$f(x) = 2\sqrt{\sin^2x-2\sin{x}\cos{x}+1}$$ ...
0
votes
1answer
46 views

Show that f is continuous at 0, but discontinuous at c for all c/=0.

I showed that f is continuous at 0 by simply using the definition of continuity. However I have no idea how to prove the second party that f is discontinuous at c for all c/=0. Thanks
1
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0answers
66 views

Two functions such that $f$ is continuous at $0$, $g$ is discontinuous at $0$, but $f+g$ is continuous at $0$?

Can anyone give an example and justify the answer. I am not sure how to prove such a thing.
0
votes
2answers
70 views

Proving that $h$ is continuous at $0$ iff $f(0) = g(0)$?

Let $f,g \colon \mathbb{R} \to \mathbb{R}$ be continuous at $0$. Define $h \colon \mathbb{R} \to \mathbb{R}$ by $$ h(x) = \begin{cases} f(x) & x \leq 0, \\ g(x) & x > 0. ...
2
votes
3answers
90 views

Finding the $n$-th derivative of $f(x)=\log\left(\frac{1+x}{1-x}\right)$

I am trying to find the general form for the $n$-th derivative of $f(x)=\log\left(\frac{1+x}{1-x}\right)$. I have rewritten the original formula as: $\log(1+x)-\log(1-x)$ for my calculations. I ...
5
votes
3answers
90 views

What is the fastest way to find the range of functions having modulus: $f(x) = |x+3| - |x+1| - |x-1| + |x-3|$

While solving problems I saw a question in which I was supposed to find the range of a function $$f(x) = |x+3| - |x+1| - |x-1| + |x-3|$$ I know the way in which I can take different cases of $x$ ...
0
votes
1answer
33 views

Determining the kernel of a function and finding the direct image of a set

Suppose we have a function $f:Z\setminus\{0\} → Z\setminus\{0\}$ such that : $$ f(x)= \begin{cases} 3x+1 & \text{when x is odd} \\ x/2 & \text{when x is even} \end{cases} $$ How would we ...
2
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0answers
51 views

If $f(x),g(x)$ are two differentiable functions such that $f'(x)=g(x)$, $|f(x)|<1$ and $f(0)^2+g(0)^2=9$ [duplicate]

If $f(x),g(x)$ are two differentiable functions such that $f'(x)=g(x)$, $|f(x)|<1$ and $f(0)^2+g(0)^2=9$, prove that there exists a $c\in(-3,3)$ such that $g(c)\cdot g''(c)<0$ ...
0
votes
0answers
21 views

How can i show the following limit graphically?

Let $f: \left[c,d\right]\to \Bbb R$, $a\in \left(c,d\right)$ and $b>0$. If $\lim_{x\to a} f(x)=l$, then prove that $\lim_{x\to \frac{a}{b}} f(bx)=l$. In order to visualize the problem draw a ...
0
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0answers
21 views

Can a Constant in a polynomial equation have a Degree or Turning Point? [duplicate]

I'm pretty sure a constant doesn't have a Degree as the Degree is the coefficient with the highest exponent. I'm still not sure about the turning point though. Thanks.
1
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0answers
41 views

Study the variation of $\frac{\Gamma(m-n^\prime+1,n^\prime q)}{\Gamma(m-n+1,n q)}$ with $q$

We define function $g(q)$ as the following: $$\frac{\Gamma(m-n^\prime+1,n^\prime q)}{\Gamma(m-n+1,n q)},$$ where $n^\prime \ge n+1$, $n^\prime \le m$. We note that $q$, $m$, $n$ $n^\prime$ are all ...
0
votes
1answer
19 views

Order of a finite sum

How to prove that the order of $$ \sum\limits_{k=0}^{n} k^{-d/2 + 1} e^{-\frac{n}{k}} $$ is $O(n^\frac{4-d}{2})$? I would like to bound the sum by integral, but the function is not monotonic.
6
votes
4answers
63 views

If $[x+0.19] +[x+0.20] +[x+0.21] +\cdots [x+0.91] =546$ find the value of $[100x]$..

Problem : If $[x+0.19] +[x+0.20] +[x+0.21] +\cdots [x+0.91] =546$ find the value of $[100x]$ where [.] represents the greatest integer function less than equal to x. My approach : $x +1.19 = x ...
0
votes
1answer
23 views

A function g defined for all real $x>0$ satisfies $g(1)=1, g'(x^2)=x^3$ for all $x>0$ then find g(4)

Problem : A function g defined for all real $x>0$ satisfies $g(1)=1, g'(x^2)=x^3$ for all $x>0$ then find g(4) Please suggest how to tackle this getting no clue how to proceed tried to form ...
1
vote
2answers
15 views

How to draw a diagram defining a map as the composition of two other maps?

I would like to know what is the most common way of drawing a diagram to define a certain map $h: X \to Z$ as the composition of some two mappings: $f : X \to Y$ and $g: Y \to Z$.
0
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0answers
16 views

how to convert to truth table and circuit

so I have to draw the switching circuit as well as truth table for this question. I have the the circuit but I'm not sure whether it is right or wrong. but for truth table I'm stuck. i) ...
0
votes
1answer
29 views

Help me to know how to generate random number under certain distribution.

I want to generate random number base on a certain distribution. Its pdf(probability density function is $f(x)=\alpha/x$ where $x∈[x_{max},x_{min}]$, $\alpha=1/\ln(x_{max}-x_{min})$ I made a vba ...
6
votes
2answers
65 views

At least $P(m, n - 1) = {{m!}\over{(m - n+1)!}}$ surjective functions from $[m]$ to $[n]$?

How do I see that there are at least$$P(m, n - 1) = {{m!}\over{(m - n+1)!}}$$surjective functions from $[m]$ to $[n]$?
0
votes
2answers
39 views

Finding a two-variable function that is distinct from another on every open disk, with specifics.

Consider the two-variable function $$f(x, y) = \sin(x) + \cos(x) + y^2.$$ Find a two-variable function $g(x, y)$ that is distinct from $f(x, y)$ on every open disk which contains the point $(1, 2)$ ...
1
vote
0answers
48 views

Prove there is no one-to-one correspondence function $\colon X→ P(X)$

I'm stuck on the following question and would appreciate it if someone could show me how I can prove it. Let $X$ be any set and let $P(X)$ be the power set containing all subsets of $X$. Prove ...
1
vote
1answer
21 views

Equality of two multilinear forms

Take two multilinear forms $f,g$ defined on the same set $E$ such that $\forall x\in E,f(x,x,\dots,x)=g(x,x,\dots,x)$. Does that imply that the two functions are necessarily equal ? I can't seem to ...
0
votes
0answers
14 views

Continuous/interpolating alternative to order of magnitude?

Define $\operatorname{magnitude}\left(x\right) = 10 ^ { \lfloor \log_{10} x \rfloor }$ and $\operatorname{magnitude'}\left(x\right) = 10 ^ { \lfloor \log_{10} x \rceil }$ Currently I'm using this ...
2
votes
2answers
32 views

Solving a radical equation with trinomials on both sides

$$8\sqrt{a^2-4a-16}=3a^2-12a-64$$ I do know the standard procedure—square both sides, isolate square root, square again, check solutions to make sure they are real, etc. However, for a problem such ...
2
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0answers
42 views

Is it correct to say $\frac{\text{d}}{\text{d}x}[F_{X}(x)] = f_{X}(x)$ or $\frac{\text{d}}{\text{d}x}[F_{X}] = f_{X}$?

Is it correct to say $\dfrac{\text{d}}{\text{d}x}[F_{X}(x)] = f_{X}(x)$ or $\dfrac{\text{d}}{\text{d}x}[F_{X}] = f_{X}$? This notation question comes up in probability, where $F_{X}$ is a CDF of a ...
1
vote
1answer
90 views

prove that the following function is decreasing?

I am trying to prove that the following function is decreasing. \begin{align}&f(t)=\frac{1-g(t)}{\sqrt{1+e^t}}\cdot\exp\left(-\frac{te^t}{2(1-e^t)}\right)&t<0\end{align}where $ ...
4
votes
3answers
44 views
1
vote
1answer
25 views

Quadratic function that produces natural number from natural number inputs

I am currently trying to find a way to generate different (preferably quadratic) function as part of a encryption algorithm such that : ...
0
votes
1answer
54 views

Prove $(g+h)\circ f=g\circ f+ h\circ f$ [closed]

Let $g,h,f$ be functions with domains and ranges on the real numbers. I have to prove that $$(g+h)\circ f=g\circ f + h\circ f$$ It seems so simple, but I don't know where to start the proof. Maybe ...
0
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0answers
39 views

Proof Verification: Surjectivity and injectivity of a piecewise function.

The following is a function such that $f: \mathbb{R}\rightarrow \mathbb{R}.$ In this case, $f(x)$ = {$x$ when $x$ is rational, $2x$ when $x$ is irrational} Determine whether $f$ is injective, ...
2
votes
2answers
30 views

Inverses of piecewise functions.

For an example, let $f: \mathbb{R}\rightarrow \mathbb{R}, $be defined by$ f(x) = 2x $ when x is rational and $f(x) = 3x$ when x is irrational. Can it simply be concluded that the inverse is ...
0
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1answer
62 views

Is the given function injective?

Let me say it with words: For some $\alpha$, if I choose $(x,y)$, then there is a quadruple $(a,b,c,d)$ which satisfies the mapping: ...
0
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0answers
18 views

Local Maxima in two variable function

Given the following function: f(x,y) = 1,000,000*y/(x+y)-y How do I find a local maxima of the function? I understand that I should calc dx=0 and dy=0 and then ...
0
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0answers
13 views

Behavior of $J/I$ w.r.t $m_1$, $I=\int_{m_1k}^{\infty} t^{N-k}e^{-t} dt$ and $J=(m_1k)^{N-k} e^{-m_1k}-\int_{m_1k}^{\infty} \log(t) t^{N-k}e^{-t} dt$

Let us define $I=\int_{m_1k}^{\infty} t^{N-k}e^{-t} dt$ and $J=(m_1k)^{N-k} e^{-m_1k}-\int_{m_1k}^{\infty} \log(t) t^{N-k}e^{-t} dt$. We assume that $m_1 \ge 0$, $k \ge 0$ and $k \le N$. Using the ...
1
vote
1answer
24 views

Limits w/ Greatest Integer and Abs. Value Function

Find the $$\lim _{x\to 2^+}\ {\lfloor x \rfloor - 1\over\lfloor x \rfloor - |x|}$$
0
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0answers
36 views

Prove that $\lim_{x\to\infty} f(x) = \alpha$

Let $f_n\to f$, uniformly on $[0,\infty)$ and let's assume that $\lim_{x\to\infty} f_n(x) = \alpha_n$ and $\alpha_n\to \alpha$. Show that: $$\lim_{x\to\infty} f(x) = \alpha$$ Now, the proof goes ...
0
votes
1answer
45 views

Study the convergence of this series of functions: $\sum_{n=1}^{\infty }n^x\left ( \tan\frac{x^n}{n}-\sin\frac{x^n}{n} \right )$

I tried to study the convergence of this series: $$\sum_{n=1}^{\infty }n^x\left ( \tan\frac{x^n}{n}-\sin\frac{x^n}{n} \right )$$ I started to study the pointwise convergence with the limit for $n ...
2
votes
0answers
24 views

Differentiability of Parameterized Inverse

Suppose I have a function $f(x,y)$ that is continuously differentiable and that for each $x$, the function $f(x,\cdot)$ is strictly increasing. Let $f_x^{-1}(\cdot)$ be the inverse function of ...
1
vote
1answer
31 views

Polar to Cartesian: r = 3 + sin(theta/2)

I am asked to convert the following polar function to cartesian: $$r = 3 + sin(\theta/2)$$ I would be able to do it if it weren't for the fraction. I have already tried substituting the identity ...
2
votes
2answers
51 views

Limit of real function, theoretical exercise

I'm a freshman in mathematics and this is my exercise: Prove that for function $f:\langle -a,a\rangle\setminus\{0\}\longrightarrow\langle0,+\infty\rangle$ so that is $$\lim_{x\to 0} ...
1
vote
2answers
66 views

Continuity of Popcorn Function (Thomae's Function)

I have to prove that the function $f:]0,1] \rightarrow \Bbb R$ : $$ f(x) = \begin{cases} \frac1q, & \text{if $x \in \Bbb Q$ with $ x=\frac{p}q$ for $p,q \in \Bbb N$ coprime} \\ 0, & \text{if ...
2
votes
2answers
63 views

Sin(x): surjective and non-surjective with different codomain?

Statement that $\operatorname{sin}(x)$ not surjective with codomain $\mathbb R$ and surjective with codomain $[-1,1]$ found here: Non-surjective: $\mathbb{R}\rightarrow\mathbb{R}: ...
1
vote
1answer
33 views

Surjective function from a countable set

In Lang "Real and Functional analysis" is demonstrated that given a countable set $A$ and a function $f: A \rightarrow B$ which is surjective on $B$, then $B$ is finite or countable. Proof: Consider ...
0
votes
1answer
40 views

How to calculate $f'(x)$ when the function has $ x^ {\gt 2}$?

I have this function: $$f(x) = |x^5 + 3x -1|$$ I need to calculate $f'(0)$ ? I am having difficulties because I don't know how to solve a function with $x^5$ ! I think this is a tricky ...