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

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1answer
124 views

Applications for holomorphic functions?

Could anyone give me an insight into practical applications of holomorphic functions (I am using the term in the way in which it is related to Riemann's work)?
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1answer
48 views

Uniform continuity of two functions

Investigate uniform continuity of the following functions: $$a) \ f(x)=\frac{1}{x} \\ b) \ f(x)=\cos \frac{1}{x}$$ How to deal with such questions, i have little knowledge about that topic thus i ...
0
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1answer
30 views

Tell if a series converges uniformly

Let $f(x) = \sum_{n=1}^\infty \frac{-2x}{(x^2+n^2)^2}$. Check if $f_n(x)$ converges to a continuous function. So I've seen a solution that uses the fact that if $f(x)$ converges uniformly and $f_n(x)...
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1answer
21 views

Showing a function to be a norm

I want to prove or disprove that $\parallel (x,y)\parallel=\sqrt{\frac{x^2}{9}+\frac{y^2}{4}}$ is a norm on $\mathbb{R^2}$. Since $\{(x,y):\parallel(x,y)\parallel\leq1\}$ is a convex set, $\parallel\...
0
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1answer
27 views

$f(x)$ be differentiable and have a local minimum in $x_0$, show with definition $f'_+(x_0)\ge0, f'_-(x_0)\le0, f'(x_0)=0$

Let $f(x)$ be differentiable in $x_0$, $x_0$ is a local minimum, prove with the definition that $f'_+(x_0)\ge0, f'_-(x_0)\le0, f'(x_0)=0$. I get that $f$ is decreasing from the left and increasing ...
1
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1answer
117 views

The maximum notation as regards the absolute value?

We know that $\max(\textbf{A})$ gives the maximum element of the array $\textbf{A}$. What is the notation, or a short formula, if we seek the element having the largest absolute value? e.g., $\textbf{...
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3answers
63 views

Size of function spaces

For example, how big is the space $ C^k(\mathbb{R},\mathbb{R}) $ ? How much is, say, $ C^0 $ larger than $ C^1$ ? How can one figure out ?
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2answers
71 views

What is the behavior of these functions linear or logarithmic or neither?

Please consider the following functions $F$ and $G$. \begin{align*} F(K) = \log_2 \left(\frac{( 2\sqrt{K-1}+K-2)^2}{(\sqrt{K-1}-1)^2}\right)+(K-1) \log_2(2) \end{align*} and the function \begin{...
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1answer
390 views

How to sketch the level curves of $f(x,y) = x^2 - y^2$

I've been practising functions of several variables for college and I've been working with circles all the time $(x^2 + y^2)$, however, I still can't figure out how to solve non circular shapes, as ...
3
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1answer
40 views

How to prove that $f=id_A$?

We have a surjective function $f:A\rightarrow A$ and we know that $f\circ f=f$. How do I prove that $f=id_A$?
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1answer
65 views

Every continuous function $f:[0,1]\rightarrow \mathbb{R}$ is bounded above [closed]

How do I prove, using the Bolzano Weierstrass Theorem, that every continuous function $f:[0,1]\rightarrow \mathbb{R}$ is bounded above? The Bolzano Weierstrass Theorem states that each bounded ...
2
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1answer
91 views

Finding inverse of $f(x) =\frac{\ln(x)}{x}$

How do you find the inverse of the following function $$f(x) = \frac{\ln (x)}{x}$$ What looked like a simple question made my head hurt during exam.
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1answer
40 views

Variations of a sinc function

Consider the following function \begin{align} G(\omega_t) = \frac{1}{N_t}\text{exp} \bigg( j \pi \Delta_t \omega_t (N_t-1)\bigg) \frac{\sin (\pi N_t \Delta_t \omega_t)}{\sin (\pi\Delta_t \omega_t)}...
1
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1answer
49 views

If $\int_{0}^{\pi}t^{n}f(t)dt=0$ for all $n\in\{0\}\cup \mathbb N$ then prove that $f \equiv0$

Let, $f:[0,\pi ]\to \mathbb R$ be a continuous function such that $f(0)=0$. If $\int_{0}^{\pi}t^{n}f(t)dt=0$ for all $n\in\{0\}\cup \mathbb N$ then prove that $f \equiv0$. Since, $f$ is ...
3
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5answers
81 views

Example $g\circ f=id_A$ but $f\circ g\neq id_B$

Two functions $f:A\rightarrow B$ and $g:B\rightarrow A$. Can someone give me an example where $g\circ f=id_A$ but $f\circ g\neq id_B$?
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2answers
48 views

Is there a non-ambiguous name for the “square of a function”?

Given a function $f$, I want to refer to $f \circ f$ other than by a formula. Is there any name for this other than square of $f$, which has the problem of being ambiguous? In analogy to the ...
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0answers
29 views

Trying to extrapolate function (some kind of $x^3$ shape)

I would need to get a generic function $f(x)$, with variable $m < z < M$ from this characteristics: $f(m) = 0$ $f(z) = 50$ $f(M) = 100$ $f'(z) = 0$ $f'$ is a increasing function with $[m,M]$ ...
0
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1answer
60 views

Find range of values for function to be onto

Suppose $f:\mathbb R\rightarrow\mathbb R$ where $f(x)=\dfrac{ax^2+6x-8}{a+6x-8x^2}$ Find the range of values of $a$ for which $f$ is onto. I tried many thing like assuming it to be $y$ then the $D\...
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1answer
450 views

Small question: Name for the x of function f such that f(x)=x?

Background When doing maths and chemistry problems, I often came across things like $$x-\frac{x}{2}=\frac{x}{2}$$ It might seems trivial, but I found that it is often the presence of expressions like ...
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0answers
28 views

What is the domain and image of the composition of mappers in a manifold

I was trying to understand the following: which I got from: http://www.mit.edu/~9.520/fall14/slides/class14/class14_manifold.pdf I was wondering, why the domain was: $$ \alpha(U_{\alpha} \cap U_{...
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0answers
38 views

Interpretation of an integral of a function $f$

When we think of a Riemann integral, it is usually defined as $\lim_{\Delta x_{k}\rightarrow 0}\sum_{k = 1}^{n}f(x_{k}^{*})\Delta x_{k} = \int_{a}^{b}f(x)~dx$. This means that $f(x)$ should be ...
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1answer
77 views

Defining the differentiation operator

The differentiation operator is the function $\frac{\mathrm{d} }{\mathrm{d} x}: f \mapsto f'$. My question is, does the operator really take an entire function $f$ as an argument? For example, when ...
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1answer
48 views

Function as onto

I have a doubt that if a function is one-to-one then it will also be onto. If a function $f(x)$ is defined such that $f: \mathbb{R} \rightarrow \mathbb{R}$ then if the function is many to one then ...
4
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1answer
42 views

Showing that f,g are invertible if $A$ is a finite set and $f,g: A\to A$ such that $f\circ g$ is invertible

Let $A$ be a finite set and $f,g: A\to A$ such that $f\circ g$ is invertible. Prove f,g are invertible. Prove that if $A$ is an infinite set, it doesn't mean that f,g are invertible. I ...
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3answers
118 views

Handling division by zero axiomatically

Suppose we define the multiplicative inverse function on real numbers as follows: $\forall{x \in \mathbb{R}}(x \neq 0 \implies x \times \frac{1}{x} = 1) $. Consider this truth table. \begin{array} {...
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0answers
31 views

Finding out if a function is invertible: $f,g:\mathbb N\to \mathbb N$, $g(x)=2x$ and $f$ with cases

Let $f,g:\mathbb N\to \mathbb N$ such that $g(x)=2x$ and $f(x)=\begin{cases}\frac x 2 &, x\in\mathbb N_{even}\\ x+9 &,x\in\mathbb N_{odd}\end{cases}$ ...
2
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1answer
81 views

Recursively enumerable sets are domain of partial recursive functions

My definition of recursively enumerable set is that it is the language recognized by some Turing machine. I want to show that this definition is equivalent to "a r.e. set is the domain of some ...
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1answer
51 views

How do I prove that the only possible function is $exp$?

Let´s say we have a differentiable function $f : \mathbb{R} -> \mathbb{R}$ with $f' = f$ and $f(0) = 1$ . How do I show that the only possible function for this to work $f = exp$ ? ...
1
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2answers
309 views

Number of one -to-one functions

Let $A = \{1, 2, 3, 4\}$ and $B = \{a, b, c, d, e\}$. what is the number of functions from $A$ to $B$ are either one-to-one or map the element $1$ to $c$? My answer is $166$, but I'm not really sure ...
0
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1answer
131 views

Can every function be a composite to itself and how to know if a composite between two functions is defined?

Can every function be a composite to itself? like we have $f:A\to B$ is $f \circ f$ always defined? Can we say that if $f$ is a injection/surjection/bijection then so is $f\circ f$? Also, how do ...
3
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4answers
48 views

Double branch $\sqrt x$ or square function turned 90°?

I have this idea for a graph but don't know what function could describe it better. The idea is something like the "squared" function turned $90$ degrees to the right, so that possible values for $x$ ...
5
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2answers
157 views

Math Contest Question with Polynomials

Prove that there does not exist a polynomial f(x) with integer coefficients for which f(2008) = 0 and f(2010) = 1867. This is a question from CMOQR (Qualifier for Canadian Math Olympiad , not the ...
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0answers
26 views

Is there a name/notation for coordinate-wise identical function?

Let's define $g: \mathbb{U}^n \rightarrow \mathbb{V}^n$ where $\mathbb{U}$ and $\mathbb{V}$ are arbitrary sets as $$g(u) = \left[f(u_1), f(u_2), \ldots, f(u_n) \right]^T$$ for some $f: \mathbb{U} \...
1
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1answer
56 views

Find the function continuous or discontinuous

$\sum_{n=1}^∞ $ $(x+2)^n \over n! + x^2$ , Interval = [1,2] Is this function continuous in that interval ? I tried but the factorials are troubles.
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4answers
65 views

Writing a piecewise function for $f(x) = \mid x+3\mid -\mid x-1\mid $

I am wanting to write a piecewise function for the following: $$f(x)= \mid x+3\mid -\mid x-1\mid $$ I know how to write piecewise functions for functions that have a single set of absolute ...
5
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1answer
94 views

Graphing a Piecewise Function

I graphed this function below. I want to make sure I am graphing piecewise functions such as this one correctly.
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1answer
89 views

Inverse function for $y=\lfloor x\rfloor+x$

Find the inverse function of the following function: $y=\lfloor x\rfloor +x$ I have tried writing down $x$ as $\lfloor x\rfloor +\{x\}$ but didn't get anywhere with that. A proper hint would ...
3
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1answer
351 views

Exponential function given two points

I am trying to find an exponential function satisfying two points (having base "exp"). After some search, I couldn't find something relative (the most relevant was that https://www.youtube.com/watch?v=...
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2answers
70 views

Is $f(x) =|x| - 3$ even, odd, or neither?

$$f(x)=|x|-3$$ Is the function above odd, even or neither? I know that a function is even if $f(x) = f(-x)$: $$f(-x) = |(-x)| - 3$$ $$f(-x) = x-3$$ Does this mean that the function is even? ...
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1answer
20 views

How to resemble hyperbolic trigonometric functions (HTF) from normal trigonometric functions(NTF)??

There are many properties of HTF similliar but little different than NTF, Is there some pattern or rule that makes me, who only knows NTF, able to get HTF from direct resemblence to NTF?
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5answers
1k views

Why does notation for functions seem to be abused and ambiguous?

I really need to clear up a few things about function notation; I can't seem to grasp how to interpret it. As of right now, I know that a function is roughly a mapping between a set $X$ and a set $Y$, ...
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0answers
45 views

Name for a nowhere constant function?

Is there a pithy name for a function $f : \mathbb{R}^n \rightarrow \mathbb{R}^m$ such that there is no non-degenerate interval $I \subseteq \mathbb{R}^n$ such that $f$ is constant on $I$ (by '$f$ is ...
3
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3answers
54 views

Multiple variables calculus: condition for $f$ to be continuous using curves

Prove $f:\mathbb{R}^n\to\mathbb{R}$ is continuous iff for every curve, $\gamma:[a,b]\to\mathbb{R}^n: f\circ \gamma :\mathbb{R}\to\mathbb{R}$ is continuous. $(\Rightarrow)$ is trivial. $(\Leftarrow)...
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4answers
95 views

Proving a set is uncountable. [duplicate]

I need to prove that the set of all functions $\mathcal{F}:\mathbb{N}\rightarrow \left \{ 0,1 \right \}$ is uncountable. I'm not too sure at all how to do this. My initial idea was to try and show ...
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2answers
3k views

Is the function $f(x) = tan(x)$ odd, even, or neither?

Is the function $f(x) = \tan(x)$ odd, even, or neither? Here is what I have so far: I know the function is not even because $f(x) ≠ f(-x):$ $$f(-x) = \tan(-x)$$ $$\tan(-x) ≠ \tan(x)$$ Now I want ...
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1answer
46 views

Prove that f(x,y) is not continuous for any a element of R

Given function $$ f(x,y) = \begin{cases} 3xy/(x^2 + y^2) & (x,y) \neq (0,0) \\ a & (x,y) = (0,0) \end{cases}$$ prove there exists no $a \in \Bbb R$ ...
0
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1answer
51 views

How to prove $(2^{-1/y}(1-x)+x)^{-y}$ is increasing in $y$, when $x,y \in (0,1)$.

As the title suggests, how to prove $(2^{-1/y}(1-x)+x)^{-y}$ is increasing in $y$ when $x,y \in (0,1)$?
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4answers
61 views

limit of the function $x \to 1$

Why $$\lim_{x\rightarrow1} \frac{ e^{ \frac{1}{\pi} \ln x } -1} {\frac{1}{\pi} \ln x }=1$$ I do not get it at all... I do not want explanation that would involve use of l'hospital rule
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2answers
42 views

Optimization problem of two variable

Find two numbers $a$ and $b$ with $a \leq b$ such that $\int_a^b (6-x-x^2)dx$ has the largest value.
3
votes
4answers
348 views

Find a path of a particle

I'm stuck with the following question: Let P be a particle at point $(1,2)$ on the surface $z=x^2y^2$. At $t=0$ the particle is left and moves freely. Find the path that the particle passes during ...