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

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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 ...
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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.
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1answer
31 views

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

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 ...
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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 ...
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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 ...
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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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2answers
110 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 ...
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2answers
31 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 ...
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1answer
29 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},& ...
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0answers
53 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. ...
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0answers
14 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, ...
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0answers
50 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 ...
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3answers
30 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 ...
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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 ...
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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 ...
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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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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?
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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 ...
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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 $$f(x)<a\in I$$ ...
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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 ...
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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 ...
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1answer
53 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 ...
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2answers
34 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} + + ...
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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.. ...
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2answers
40 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 ?
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2answers
61 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.
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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?
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2answers
35 views

Functions - finding the domain

Question: Consider the function: $$f(x) = \log(2x + 1) - \log(x - 3)$$ What will be the domain of this function? I used two approaches to solve this question. Both approaches got me different ...
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1answer
31 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 ...
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1answer
23 views

Total number of distinct solution produced by polynomial

I have a function $F(x,y) = ax + by$ where $x,y$ belongs to range $[1..10^{10}]$ and $a$ and $b$ are constants, all are integers. How many distinct values can be produced by this function, please give ...
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1answer
37 views

If the cos of 27 is 0.89, how much is the csc of 27

Hey guys for my trig class we're viewing trigonometric functions and their properties. So far I have understood but I came across this problem and can't seem to solve it: Given the approximation cos ...
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1answer
20 views

Function that maps a rational number to its numerator and denominator

Question: Is there a simple way to represent a function $f:\mathbb Q\to \mathbb Z^2$ that maps a rational number in lowest terms $r=\frac ab$ to the ordered pair of its numerator and denominator ...
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1answer
33 views

Understanding relation between vector valued function and function objective in an multi objective optimization problem

I try to understand the relation between "vector-valued function" and "function objective" as used in optimization problem. I understand that objective function in a multi-objective problem can be ...
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1answer
31 views

What is the highest order of derivative of this function $f(x) = x^5\sin(\frac{1}{x}) $ at $x=0$?

The function is defined as $f(x) = x^5\sin(\frac{1}{x}) \quad \text{for} \quad x\neq 0 \quad $ and $f(x) = 0$ for $x=0$. I can't tell by just looking at the plot. I think there might be a theorem I ...
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0answers
19 views

A question about a notation used in the Folland Real Analysis

This is the exercise 11 in the Folland Real Analysis. Could anyone tell me what it means by f(x,・) and f(・,y)? I have never seen such notations before...
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0answers
223 views

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 ...
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5answers
71 views

Find the limit of the function [closed]

Let we have the following function $$F(x)=\frac{x^x-x}{\ln(x)-x+1}$$ Find $$\lim_{x \to 1}\frac{x^x-x}{\ln(x)-x+1}$$
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5answers
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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 ...
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1answer
29 views

From a point-wise to a linear piece-wise definition

Consider the following definition: $$f(x, \alpha) = \left\{\begin{array}{cl}A(x) & \alpha = 0\\ B(x) & \alpha = 0.5 \\ C(x) & \alpha=1 \end{array}\right.$$ where we always have $A(x) < ...
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0answers
25 views

Applying implicit function theorem to function with derivative

This may be a very peculiar question, or I may even be on the completely wrong track, so I apologize in advance for obvious errors. I am trying to apply the implicit function theorem in an ...
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1answer
26 views

Periodic functions proof

I need some help here. Let $f$ be a $2\pi$-periodic function, and define for an arbitrary $k\in\mathbb N$ a function $g(x) = f(kx)$. Show that $g$ is also $2\pi$-periodic. What I've done: $$ g(x) ...
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2answers
17 views

Defining domain in complex plane

I am asked to define the domain for the following given that $z=x+iy$: $a) \quad f(z) = \dfrac 1 {z^2 + 1}$ $b) \quad f(z) = \dfrac 1 {1 - |z|^2}$ How would this be different from a normal domain ...
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2answers
59 views

Can we give a bound on any associative function?

We say that $f:[1,\infty)^2\to[1,\infty)$ is associative if $$f(f(a,b),c)=f(a,f(b,c))$$ And symmetric if $$f(a,b)=f(b,a)$$ e.g. the arithmetic operations '+' and '$\cdot$' are associative and ...
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1answer
9 views

question about vacuous truth and function

I'm confusing about vacuous truth. Let $f: \mathbb{N} \rightarrow \mathbb{N}$ where $f(n)=2n$. we can calculate function values if $n$ belongs to domain. but what if it does not? The value of ...
3
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3answers
208 views

I cannot make the mental leap from a vector to a function!

In my linear algebra book, it says that a vector is linearly independent if $\vec V = c1*\vec T_1 + c2*\vec T_2$ And then it goes on to say that $y(t) = c1 * e^{-at} + c2*e^{-bt}$ is linearly ...
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0answers
9 views

Tigher bounds on concavity of log

Is there a tighter upper and lower bound on concavity of $\log(.)$ function? It is very well known that $\log(\sum_{i}p_ix_i) - \sum_{i}p_i\log(x_i) \geq 0$. But are there stronger versions of this ...
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1answer
16 views

How to interpret algebraic relationship/ next step to take to prove function is onto?

This is a problem from Discrete Mathematics and its Applications Book's definition on bijection Book's definition on onto Book's definition on one to one I am trying to do problem 23D. Here ...
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3answers
43 views

Solve for $y$ in $x=\sqrt{(y-1)/(y+1)}$

I always struggle with this: Express $y$ in terms of $x$ where $$x = \sqrt\frac{y-1}{y+1}$$ I know to square both sides and get $x^2 = \frac{y-1}{y+1}$ Then I'm thinking multiply both sides ...
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2answers
70 views

Proof of $\cos(y)$ and $\sin(y)$ using $e^{iy}$

I need to use that $e^{iy} = \cos y + i \sin y$ (for $y \in \mathbb{R}$) to prove that $$\cos y = \frac{e^{iy}+e^{-iy}}{2}$$ and $$\sin y = \frac{e^{iy}-e^{-iy}}{2i}$$ I'm really clueless, any ...
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0answers
27 views

How to find an injuctive function for not divisible by 7 but divisible by 5?

This is based off my other question - How to write a function to express not divisible by 3? This is a problem from Discrete Mathematics and its applications I am currently on 4B. Here is my work ...