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

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0
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0answers
25 views

Continuous and additive function is linear [duplicate]

Let $f:\mathbb{R}\to \mathbb{R}$ be a continuous function and $f(x+y)=f(x)+f(y)$, show that $f(x)=kx$, $k\in \mathbb{R}$. I tried to define $g(x)=f(x)-kx$ and $g(0)=0 $ but don't know how to ...
1
vote
1answer
29 views

Functions that preserve equivalence relations

Quick question: Let $X$ and $Y$ bet two sets and $\sim$ an equivalence relation on $X$. I was wondering what it means to say that a function $f$: $X\to Y$ 'preserves' $\sim$ in this case. Does it ...
0
votes
1answer
10 views

Why is the CT system $y(t)=x(2t)$ invertible but its DT counterpart non-invertible?

Just for clarity, a system is invertible if distinct inputs lead to distinct outputs. That said, I have two systems, a continuous time system and a discrete time system: (1) $y(t) = x(2t)$ ...
2
votes
1answer
19 views

How find $\min_{a\in\mathbb R}f\left(a \right)$ for $f(a )=\max_{1\le k\le n}\left|x_k+ay_k\right|$?

Let them be given points in the plane $P\left(x_k,y_k\right)$, where $k\in \{1,...,n\}$. Let $f(a )=\max_{1\le k\le n}\left|x_k+ay_k\right|$ , where $a\in\mathbb R$ . How find $\min_{a\in\mathbb ...
0
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3answers
40 views

Help me with proof concerning functions

Problem: Let $X$ and $Y$ be non-empty sets and let $f: X \rightarrow Y$ be a function. We define $F: P(Y) \rightarrow P(X)$ by $F(B) = f^{-1}(B)$ for all $B \in P(Y)$. Proof that $F$ is injective if ...
0
votes
1answer
53 views

Was my inference regarding $u(z)=log(z)$ correct?

I have already solved the problem but would appreciate a clarification in part (b). A has initial wealth $w$ and faces a loss $l$ with known probability $\pi$. Insurance available at unit price ...
4
votes
2answers
98 views

Solving functional equation $f(x+y)+f(x-y)=2f(x)\cos y$?

How can I solve this functional equation, where $x,y$ are any real numbers and $f:\mathbb{R}\to \mathbb R$ is a function such that : $$f(x+y)+f(x-y)=2f(x)\cos y$$ I tried substituting $x=0$ to get ...
5
votes
1answer
31 views

Difficulty with Jensen's Equation.

Its easy to find all continuous function $f: \Bbb R \to \Bbb R $ satisfing the Jensen equation $$f \left( \frac{x+y}{2}\right )=\frac{f(x)+f(y)}{2}$$ But I am finding difficulty in finding all ...
0
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0answers
26 views

Proof that there is no identity to integral operation on any set of functions

The statement is: Let $f\in F, f:x\mapsto f(x)$ be a function($F$ contains sufficiently non-trivial functions). Then $\not\exists I\in F$, so that $$\int_{-\infty}^\infty If=f(0)$$ What I am implying: ...
0
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1answer
36 views

Find a Taylor series around $x=0$ [on hold]

I don't know how to find the Taylor series around $x=0$ for: $$f(x)=\frac{\tan(2x)-\arctan(4\sinh(x))}{\sin(x^{2})}$$ Thank you in advance.
1
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0answers
23 views

Convexity of matrix inverse

If $$f\colon\mathbb{R}^n\times\mathbb{R}^n\to\mathbb{R}$$ where $$f(x,y)=y^T x^{-1}y$$ and dom$(f)=\{(x,y)\ |\ x+x^T\gt 0\}$, then is $f$ convex?
0
votes
1answer
21 views

Linear Transformations of Functions

$\textbf{Problem}$ Define $f: \mathbb{R} \rightarrow \mathbb{R}$ by $f(x) = mx + b$. $\textbf{a.}$ Show that $f$ is a linear transformation when $b = 0$. $\textbf{b.}$ Find a property of linear ...
0
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0answers
32 views

The top 1% own 50% of the world's wealth - how do we turn this into a function?

This Oxfam report states that 1% of the world's richest own 50% of the wealth. But to be in the top 1% - you don't have to be a billionaire (assuming a billion is US dollar one thousand million). ...
0
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0answers
4 views

Is there a name for this type of operation on graphs of functions?

Suppose I have a function $f(x)$ that is defined on $\{0\}\cup[1/2,1]$ such that $f(0)=0$ and $f(x)=1+x$ for $1/2\leq x\leq1$. I want to define the following extension of this function that is ...
0
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2answers
39 views

The powerset of the set of natural numbers - Cantor's Theorem

It is a fact that if $A$ is any set then there is no bijection between $A$ and its powerset $P(A)$. If $A$ is finite, this is pretty clear just by looking at the sizes of $A$ and $P(A)$. But if I ...
-4
votes
3answers
66 views

Does there exist a function between arbitrary sets?

Given arbitrary sets $A$ and $B$, does there exist a function $f: A\rightarrow B$ that is injective?. Does this follow from the axioms of set theory? If yes, then which axiom?
2
votes
3answers
28 views

Finding the formula of a function based on output

This is probably something super simple, but I can't find it in my book, and I don't even know what to search for because I don't know what to call it. I'm not looking for this specific answer, but ...
2
votes
1answer
101 views

Does such a function exist: $\left|f(x) - f(x+1)\right| \ge {1\over x+1}$?

Does a function $f(x)$ exist such that $\left|f(x) - f(x+1)\right| \ge {1\over x+1}$ where the range of the function does not extend infinitely in the positive direction. If not why?
0
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1answer
9 views

Proof concerning indexed family of sets

Let $f: A \rightarrow B$ be a function. Let $I$ be a non-empty set, and let $\left\{U_i\right\}_{i \in I}$ be a family of sets indexed by $I$ such that $U_i \subset A$ for all $ i \in I$. Proof the ...
2
votes
4answers
89 views

Find the inverse function $x + \sqrt {x}$

$ Y = x + \sqrt {x} $ Hello , I want to find the inverse function of this function , I know that it's injective How to prove the $f(x) = \sqrt{x + \sqrt{x}}$ is injective. but do not know how to ...
0
votes
2answers
27 views

The range of $\frac{2^x-1}{2^x+1}$

I am trying to find the range of the function $\frac{2^x-1}{2^x+1}$. If we draw using a graph plotter we can see that the range is $-2<y<2$. To find the upper bound, I tried ...
1
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1answer
32 views

Is $f(x,y)=-xy$ neither concave nor convex?

Is $f(x,y)=-xy$ neither concave nor convex? I used the definition for first differentiable functions and determined it depends on the choice of points, hence it is neither.
1
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1answer
14 views

Composed of non differentiable functions

It will be possible to find a function $f:\mathbb{R}\rightarrow \mathbb{R}$ non-differentiable at zero such that $f\circ g$ is differentiable at zero where $g:\mathbb{R}\rightarrow \mathbb{R}$ is ...
0
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1answer
31 views

three elementary problems on limits of several variable . [on hold]

I'm learning limits of several variable new. Can anyone help me? Computing the following limits: $\lim_{(x,y)\to(0,0)}|x|^y$ $\lim_{(x,y)\to(0,0)}\sin(x/y)$ $\lim_{(x,y)\to(0,0)}x^2\cdot ...
7
votes
0answers
140 views
+100

Approximation of the exponential

Let $c>1,k\in\mathbb{N}$. Let's consider two approximations of the exponential function : The first one is the most common one $f_k(x)=\left(1+\frac{x}{c^k}\right)^{c^k}$ and the second one is ...
4
votes
4answers
75 views

Question about $x\mapsto f(x)$ notation.

I'm trying to learn this notation, but I have some questions regarding its uses: Why is a "$:$" used instead of "$=$" when defining the function, e.g. $f: x\mapsto f(x)$ isntead of $f = x\mapsto ...
0
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2answers
31 views

Properties for functions $f:[a,b] \to \mathbb R$? [closed]

Let $f:[a,b] \to \mathbb R$ be a function. Which of the followings are true: A) If $f(x)$ is continuous then it is bounded. B) If $f(x)$ is continuous then it is increasing. C) If ...
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0answers
48 views

finding the formula to a given table of values

I created a spreadsheet that i filled with values i got from a game. The values may be rounded, but they were calculated, so there has to be a formula behind. ...
1
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2answers
30 views

Discrete math functions proof

Let $\mathbb N_{\text{even}}$ be the set of all natural even numbers, and $\mathbb N_{\text{odd}}$ be the set of all natural odd numbers, the function $f:\mathcal P(\mathbb N)\to \mathcal P(\mathbb ...
0
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0answers
11 views

Specific utility (error) function for machine learning

I need a differentiable analog of following piecewise-defined function for machine learning application: $E=E(x,y)$ when $y=1$, $E=1/(x+1)$ when $y=-1$, $E=-1/(x-1)$ $y\in \{-1,1\}$ (two values, ...
0
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1answer
28 views

how to solve piecewise function? [closed]

Does anyone know how to find the domain of function g? Did $1-x$ will effect the interval of $g$? Given $$f(x)=\begin{cases}\frac{1}{2},& x\in\left[0,\frac{1}{2}\right]\\2x-\frac{1}{2},& ...
3
votes
0answers
49 views

Real analytic function with radius of convergence 1 at non-negative integers

So, as the title states, the problem I was confronted with was to find a real-valued everywhere analytic function $$f:\mathbb{R}\to \mathbb{R}$$ s.t. at every non-negative integer, k ...
1
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3answers
69 views

Roots of this third degree polynomial

I've got the following polynomial $$ x^3-6x^2-2x+40 $$ and I want to find its roots. The only option I see at the moment is to compute all the divisors of $40$ and their inverse, and manually check if ...
0
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1answer
51 views

When the imaginary part of a function is zero?

Let $z_k=x_k+ i y_k, x_i,y_i \in \mathbb{R}$ are the complex variables. Consider a polynomial of $z_k$ and its conjugates $f(z_1,\ldots,z_n, \bar{z}_1, \ldots,\bar{z}_n).$ Question:Is there any ...
1
vote
1answer
41 views

Are functions infinite dimensional vectors? [closed]

Are functions infinite dimensional vectors? There are a few sources on the internet that makes this claim, but they do not cite any sources which makes me feel like they are just using it as an ...
1
vote
1answer
30 views

Why is $\cos((\omega+\alpha\cos(\omega' t))t)$ the wrong model for frequency modulation?

So I was trying to program vibrato, or freqency modulation, naively using the model: $$\cos((\omega + \alpha\cos(\omega' t))t)$$ Where $\alpha \lt \omega$ and $\omega' \ll \omega$. For practical ...
1
vote
1answer
47 views

A question about a linear algebra proof [closed]

If $f(x)$ is a function with domain $R$ such that for all real $a, x$ it is $f(ax) = af(x)$ then there exists a real number $b$ such that $f(x) = bx$ for all $x.$ How to prove this statement?
0
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1answer
28 views

Draw $-dU(x)/dx$ for $U(x)$

It's been a little while since I've done any problems like this, but I just wanted to make sure I'm on the right track. Updated attempt:
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0answers
31 views

How to display one to one correspondence for all bit strings not containing the bit O?

This is a problem from Discrete Mathematics and its Applications From the onset I saw that this set was countable was that you could physically count these out - 1, 11, 111, 1111 and perhaps ...
1
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1answer
16 views

Prove that, if $f(x)<a$ for $x\in I_1$ and $g(x)<a$ for $x\in I_2$, then $\sup I_1\geq\sup I_2$.

Let us consider two functions $$f:\mathbb R\rightarrow I$$ $$g:\mathbb R\rightarrow I$$ and I a subset of $\mathbb R$. Let $f(x)\leq g(x)$, $\forall x\in \mathbb R$. Prove that, if $$f(x)<a\in I$$ ...
3
votes
2answers
33 views

Another question about $x_0$ in the Taylor series

When we talk about Taylor series, we say it's around point $x_0$. It's in the Taylor series formula: $$f(x) = f(x_0) + (x-x_0)f'(x_0) + \frac{(x-x_0)^2f''(x_0)}{2} + \frac{(x-x_0)^3f'''(x_0)}{6} + + ...
1
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0answers
133 views

How many complex functions reduce to a given x-y function?

A 2D or x-y coordinate function has a complex analog, which is formed by replacing x with with the complex variable z. That function can then be separated into real and imaginary parts. Graphing the ...
0
votes
1answer
21 views

How to find the domain of this trig function?

f(x)=sqrt(tan(2x+π)) Allright, so i know you cannot have a number less than zero under the square root sign and that tangent cannot equal π/2+nπ. So should i try to find the domain of the tan ...
0
votes
2answers
39 views

To find inverse of function [closed]

Given $ f(x) = \begin{cases} 2x, & \text{if $x\in[0,1]$} \\ 8 - 2x, & \text{if $x\in [2,3)$} \end{cases} $ Then how to find inverse of f ?
21
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5answers
3k views

Why can a circle be described by an equation but not by a function?

In high-school math functions always looked to me just like glorified equations. The only time I saw a meaningful difference was when we covered the equation of a circle and I realized that an ...
1
vote
1answer
32 views

Find the max and the min of the function

I have this function: $$g(t)=t^2+\cos(2t)-\cos(t)$$ $$0\le t\le2\pi$$ I made the derivative:$$g(t)'=2t-2\sin(2t)+\sin(t)$$ And except for the obvious solution $t=0$ I'm not able to find the other.. ...
1
vote
2answers
55 views

Find the range of $ x-\sqrt{4-x^2}$

$Y=x-\sqrt{4-x^2}$. How to find these types of functions' range? I just know that the answer is $R=\{y\in\mathbb{R}\mid-2\sqrt{2}\leq y\leq 2\}$, but I have no idea how to find it step by step.
0
votes
1answer
25 views

Smooth saturation function

I need a function similar to $$Saturation(x)=min(max(x, -1), 1)$$ except for I need it to be smooth with no jump in its derivatives. It seems $arctan$ is not a good candidate since I need it to keep ...
3
votes
3answers
177 views

Approximate $|x|$ with a smooth function

I am trying to get the derivative of $|x|$, and I want that derivative function, say $g(x)$, to be a function of x. So it really needs the |x| to be smooth (ex. $x^2$); I am wondering what is the ...
0
votes
1answer
25 views

Can the domain and co-domain be the same set? Is this a function?

Let $A$ denote the set of all real numbers. Let $B$ denote the same set as $A$. Let $f$ be the function that, to each number in $A$ assigns the cube of the number. Is $f$ a function?