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Questions tagged [functions]

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

0
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0answers
10 views

Infering the shape of function from its components

Is it legitimate to say that, if all of the components of a function are linear (say of a vector function), then so is this function? How far can we take similair arguments without rigorous proof?
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votes
2answers
21 views

Prove formally that multivariate function has global extrema

Consider the following function: $f(x_1, \dots, x_n) = p_1 \log x_1 + \dots + p_n\log x_n$ subject to the constraint that $\sum_i p_i = \sum_i x_i = 1$. It is also known that $p_i \in [0,1]$ and $x_i \...
-5
votes
0answers
30 views

If $f(b)=\max\left|\sin x+\frac{2}{\sin x+3} +b\right|$, what is $\min f(b)$?

For any real number $b$ let $f(b)$ denote the maximum of: $$ \left|\sin x+\frac{2}{\sin x+3} +b\right|$$ for all $x$ belongs to $\mathbb R$. Then the minimum value of $f(b)$ for all $b$ belongs to $\...
0
votes
0answers
17 views

Longest function for the not each other dominating points

I stumbled over this little exercise on StackOverflow where the poster asks for an algorithm to find the biggest subset of a point cloud which contains only points that do not dominate each other. $...
2
votes
3answers
27 views

If $f(x)=\lim_{t\to\infty}{\frac{(1+\sin{\pi x})^t-1}{(1+\sin{\pi x})^t+1}}$, then range of $f(x)$ is?

If $$f(x)=\lim_{t\to\infty}{\frac{(1+\sin{\pi x})^t-1}{(1+\sin{\pi x})^t+1}}$$ Then range of $f(x)$ is? My Attempt: I was able to conclude that when, $$\sin{\pi x}\to0^+, f(x)\to1$$ $$\sin{\...
0
votes
0answers
8 views

Name and conditions for the property that $argmax_x f(x,y)$ doesn't depend on $y$ and vice versa

Are there necessary and sufficient conditions (or canonical sufficient conditions) for the property that $argmax_x f(x,y)$ doesn't depend on $y$, and $argmax_y f(x,y)$ doesn't depend on $x$? And is ...
0
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3answers
35 views

Is there a difference between these two notations, $f(x)$ and $f((x))$?

I'm curious, are these two notations same or different? If they are different, can anybody point out difference? That is, is $f(x)$ and $f((x))$ the same thing? Similarly, what about $f(x,y)$ and $f((...
1
vote
0answers
44 views

Convergence or divergence of the improper integral [on hold]

We have an arbitrary function $u(x)$. It is known that $u(x)$ continuous and monotonically increasing in $[0,\rho]$, $u(0)>0$, $1-\delta\,u(\rho)=0\,(\delta>0)$, $1-\delta\,u(0)>0$ and $u'(\...
0
votes
1answer
52 views

$f: \mathbb R^2 \to \mathbb R$ be $f(x,y) = x^{[y]}$. How can we define this function at $(0,0)$?

For any $y \in \mathbb R$, let $[y]$ denotes the greatest integer less than or equal to $y$. Define $f: \mathbb R^2 \to \mathbb R$ be $f(x,y) = x^{[y]}$. Then there are four options from which any ...
2
votes
3answers
51 views

Is there a bijection $\big\{x:\mathbb{N}\to\{0,1\}:\{i\in\mathbb{N}:x(i)=1\}\text{ is finite}\big\}\to\big\{x:\mathbb{N}\to\{0,1\}\big\}$? [duplicate]

Can we use Cantor's theorem to figure this out? Intuitively the domain seems to be a subset of the codomain, but where do powersets come in? Thanks in advance!
4
votes
4answers
25 views

$F(x)$ is a periodic function with petiod $k$

If $f$ be a periodic function with period $k$ and $f(-x)=-f(x)$ in $\bigg[-\frac{k}{2}\;,\frac{k}{2}\bigg]$. Then prove that $\displaystyle \int^{x}_{a}f(t)dt$ is a periodic function with period $k$ ...
1
vote
3answers
59 views

How to prove the Bijection of an Interval $(-1,1)$ to $\mathbb{R}$?

How can I prove that this image is bijective? $$f: (-1,1) \longrightarrow \mathbb{R}, \quad x \longmapsto \frac{x}{1-x^2} $$ is bijective without the use of the Steepness of the slope?
1
vote
2answers
32 views

Transforming $y=f(x)$ to $|y|=f(x)$

I am learning transformation of graphs and I have studied some of the standard rules. But this part has me stuck. How do I transform a graph from $$y=f(x)$$ to $$|y|=f(x)$$. For example, I was trying ...
1
vote
5answers
46 views

What does $f(x,y)$ mean?

I know from the chapter "functions" that $f(x)$ is a function of $x$ and to roughly put it, it maps $x$ values to another set called co-domain where all the $y$ values are. But I also sometimes see $...
0
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3answers
30 views

A curve has equation $y=5-2x+x^2$ and a line has equation $y=2x+k$, where $k$ is a constant. Answer the question in the body of text below.

For one value of $k$, the line intersects the curve at two points, A and B, where the co-ordinates of A are $(-2,13)$. Can you help me find the co-ordinates of B? I don't remember how curve co-...
0
votes
1answer
21 views

Theorem stating the connection of limits of sequences and functions.

Following is the theorem I have been given for regarding the connection between limits of sequences and functions, in order to help decide whether a limit for a function does exist or not: Theorem: ...
-3
votes
0answers
29 views

A function that behaves like the floor function and is continuous

I'm looking for a function that behaves like the floor function for integer/non-integer values given to it... to be more specific, I'm looking for a function that'd perform a set of operations to give ...
2
votes
2answers
104 views

Range of a rational function with radicals

Find the range of the function $$\frac{6}{5\sqrt{x^2-10x+29} - 2}$$ I tried using inverses, but the equation got super messy and I dont think its a good method for this problem. $\frac{6}{5\sqrt{x^...
0
votes
2answers
17 views

Writing equations that represent 3D and 2D objects

I want to represent a speed-breaker that I find in road in the form of an equation. Let's call it 3D speed-breaker. 3-Dimensions The following is the speed-breaker that I can draw on a piece of a ...
2
votes
3answers
70 views

Let $f(x^4+8x)=24x$ where $x < 0$, find $f^\prime(0)$

Let $f(x^4+8x)=24x$ where $x < 0$, which of the following equals to $f^\prime(0)$ ? $\text{a)}~1~~~~\text{b)}~-1~~~~\text{c)}~2~~~~\text{d)}~ -2$ This question was asked in high school ...
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0answers
49 views

Prove that $b_\infty(x) = \sum_{i=0}^{\infty} x(i) \cdot 2^i$ is a bijection.

Here's the problem statement: Let $X = \{ x:\mathbb{N}\to\{0,1\}:\{i\in\mathbb{N}: x(i) = 1\}$ is finite $\}$. The function $b_\infty: X \to \mathbb{N}$ defined by $b_\infty(x)=\sum_{i=0}^{...
1
vote
1answer
26 views

function inverses with exponential, why is $x = 0$?

I'm trying to learn how to use function inverses to solve equations. In an exercise I know that the solution is $0$. I am unable to reach zero myself. Solve for $x$ in the equation $g(x) = 1$, given $...
1
vote
1answer
85 views

Given that $f(x^2+x+1)=f(x^2-x+1)$ for all $x$, is $f(x)$ periodic?

Given that $f(x)$ is a function defined on $\mathbb{R}$, satisfying $$f(x^2+x+1)=f(x^2-x+1)\;\;\; \forall\;\;\;x\in\mathbb{R}$$ Is $f(x)$ periodic? My Attempt: $$f((x+\frac{1}{2})^2+\frac{3}{4})=...
1
vote
3answers
59 views

Prove $\forall x,y \in \mathbb{R} :\lfloor{x+y}\rfloor=\lfloor{x}\rfloor+\lfloor{y}\rfloor∨\lfloor{x+y}\rfloor=\lfloor{x}\rfloor+\lfloor{y}\rfloor+1$

Prove $∀x,y\ (x,y\in \mathbb{R}: \lfloor{x+y}\rfloor=\lfloor{x}\rfloor+\lfloor{y}\rfloor∨\lfloor{x+y}\rfloor=\lfloor{x}\rfloor+\lfloor{y}\rfloor+1)$ So, I let $\lfloor{x}\rfloor=m ≡ m≤x<m+1$ $...
0
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3answers
55 views

Prove that $g(x) = {x \over (1-|x|) } $ is a bijection.

I'm struggling with the following homework assignment. Let $X = \{x \in \mathbb{R}: -1 < x < 1\}$. We define $g: X \rightarrow \mathbb{R}$ by $g(x) = \frac{x}{1-|x|}$. We must prove that $g$ ...
0
votes
1answer
48 views

Condition to an increasing function on $\mathbb{N}$ be constant [on hold]

Let $f \colon \mathbb{N} \to \mathbb{N}$ a non-decreasing function. Suppose the following assertion: $$ \exists m,n \in \mathbb{N}, m > n : f(m) - f(n) > (m-n)^2$$ Can be $f$ a constant ...
0
votes
0answers
24 views

Question at the end of the chapter on Functions in Halmos's naive set theory [duplicate]

(i) $Y^\emptyset$ has exactly one element, namely $\emptyset$, whether $Y$ is empty or not, and (ii) if $X$ is not empty, then $\emptyset^X$ is empty. How do you prove these statements to be true?
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2answers
28 views

Simple variable understanding question

A function $f \colon X \to Y$ is a rule which assigns to each element $x \in X$ a unique element $y \in Y$. I don't understand what it is meant by each element $x$. If it was “all $x$” or “each $x$” ...
0
votes
2answers
35 views

Suppose that $f: \mathbb Z\to \mathbb Z$ is an onto function. Prove that $g: \mathbb Z\to \mathbb Z$ given by $g(x) = f(x) + 4$ is also onto. [on hold]

Suppose that $f: \mathbb Z\to \mathbb Z$ is an onto function. Define a function $g: \mathbb Z \to \mathbb Z$ by: for all $x\in \mathbb Z$, $g(x) = f(x) + 4$. Prove that $g$ is also onto. I have ...
0
votes
1answer
42 views

Prove the differentiability of an inverse function, by the theorem of Caratheodory.

The exact statement I am looking forward to prove is that: My try: By the theorem of Caratheodory, there exists a function $\phi:(a,b)\to\mathbb{R}$, $$f(x)-f(m)=\phi(x)(x-m)$$ continuous at $x=m$. ...
1
vote
2answers
24 views

Function For Sine Wave Between Two Exponential Cuves

What function could I use to approximate a curve with this plot? The function ideally will involve sine and have its envelope defined by different exponentials of the form $y = ae^{bx}$ and $y = ce^{...
2
votes
1answer
26 views

Term for functions that map functions to other functions

For example, let's define the "swap" function $SW(f(x,y))$ as the function that maps $f(x,y) \rightarrow f(y,x)$. I can imagine there are many such functions that have been described. Is there any ...
2
votes
3answers
40 views

Simplifying a function of $\ln(x)$

I was asked to convert the function $$\frac{1}{x \ln x \sqrt{(\ln x)^2-1}}$$ into a function in the expression $$\frac{1}{x \sqrt{f(x)}}$$ for the domain $x > e$ but I can't seem to find how I ...
-1
votes
0answers
22 views

Given same variable find two function's maximum, while one function decreasing, the other increasing trend

I am bioinformatician and i do not have a solid and deep mathematical knowledge. I have done a grid-search analysis; i got the number of protein interactions and overlap with a confirmed correlation ...
1
vote
2answers
70 views

Let $f:\Bbb{R}^n\to \Bbb{R} $ be differentiable. Prove that $f$ is linear and show that $f(0)=0$

Let $f:\Bbb{R}^n\to \Bbb{R} $ be differentiable such that $$f(\lambda x)=\lambda f(x),\;\forall\;\lambda\in\Bbb{R},\;\forall\;x\in\Bbb{R}^n.$$ Prove that $f(0)=0.$ Prove that $f$ is linear. Here's ...
8
votes
3answers
315 views

What is the domain of a division of functions?

This question is about real functions of real variables. I think that, in general, if the domain of some function $f(x)$ is A, and the domain of another function $g(x)$ is B, then the domain of $(f/g)...
4
votes
1answer
83 views

Are functions satisfying a certain inequality monotone

Let $f: (0, \infty) \rightarrow (0,\infty)$ be a continuous function which satisfies the inequality $$f(x) + f(y) \geq 2f(x+y).$$ Is $f$ necessarily monotone?
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votes
2answers
34 views

The domain of a continuous real function must be connected?

The domain of a continuous real function must be connected? For instance, the function $$ f(x) = \begin{cases} \ \ \, 1 & \text{if $x \in (1,2)$} \\ -1& \text{if $x \in (2,3)$} \end{cases} $$ ...
0
votes
1answer
35 views

Is there a function $f$ such that $f(.7x+.3y)=.5f(x)+.5f(y)$ for all $0\le x\le y\le 1$? [on hold]

Find a function $f$ such that $f(0.7x+0.3y)=0.5f(x)+0.5f(y)$ for all $0\le x\le y\le 1$? I know that a piecewise linear function can satisfy the requirement for some $x\le y$, but not all.
1
vote
5answers
44 views

When should I measure the angle of a complex number clockwise or anticlockwise?

In this diagram theta is measured anticlockwise. How would I know from which side to measure the angle? In this diagram theta is measured clockwise.
-3
votes
2answers
42 views

$f\left(x\right)=\frac{x+2}{x+3}$ find $f^{-1}$ , domain, range and functional form. [closed]

(a) Show from the definition that $f$ is one-to-one. (The horizontal line test is not sufficient.) (b) Hence find $f^{-1}$. (c) State the domain and range of both $f$ and $f^{-1}$. (d) Using the ...
1
vote
1answer
26 views

Finding the least sum of digits possible for an outcome of a function in prime numbers.

Let $f(n)=p^4-5p^2+13$ simplified as $f(n)=(p^2-{5\over2})^2+{27\over4}$ where $p$ is an odd prime. Find the least possible sum of digit of $f(n)$. My findings: After putting $p=(3n,3n+1,3n+2)$ in $...
0
votes
1answer
22 views

Signum if x tends to 0

$$\lim_{x\to0} Sgn(x)$$ What should its value be? I know $Sgn(0)=0$, but if we imply that x tends to 0, shouldn't it be an infinitesimal number close to 0, but not equal to zero, and shouldn't its ...
1
vote
1answer
37 views

2 variable Functional Equations

Suppose a function $f : R->R$ satisfies the following conditions $f(4xy) = 2y[f(x+y)+f(x-y)]$ $f(5)=3$ What is the value of $f(2015)$? I am currently stuck after $x=y$ which gives out: $f(4y^2) ...
2
votes
1answer
45 views

Why isn't this true for $x<0$?

Prove that $$\lfloor{x}\rfloor=\bigg\lfloor{\frac{x+1}{2}}\bigg\rfloor+\bigg\lfloor{\frac{x+2}{2^2}}\bigg\rfloor+\bigg\lfloor{\frac{x+2^2}{2^3}}\bigg\rfloor+ \ldots$$ $x\geq{0}$ I asked this ...
0
votes
3answers
56 views

If $f(x)$ is odd, then prove that $f'(0)=0$

If $f(x)$ is odd, then prove that $f'(0)=0$, assuming that $f'(0)$ exists. We have that $f(x)$ is odd. Therefore, $f(-x)=-f(x)\implies -f'(-x)=-f'(x)$. What I do after this?
0
votes
1answer
45 views

Proving that function space is a vector space over field

I'm reading Serge Lang's (S.L) linear algebra book. In the beginning, at function spaces section there is such a text: Let $S$ be a set and $K$ a field. By a function of $S$ into $K$ we shall ...
0
votes
0answers
43 views

Prove that $\lfloor{x}\rfloor=\bigg\lfloor{\frac{x+1}{2}}\bigg\rfloor+\bigg\lfloor{\frac{x+2}{2^2}}\bigg\rfloor+\ldots$ [duplicate]

Prove that $$\lfloor{x}\rfloor=\bigg\lfloor{\frac{x+1}{2}}\bigg\rfloor+\bigg\lfloor{\frac{x+2}{2^2}}\bigg\rfloor+\bigg\lfloor{\frac{x+2^2}{2^3}}\bigg\rfloor+ \ldots$$ $x\geq{0}$ How should I ...
0
votes
1answer
23 views

Natural domain for $f(2x)…$

There is an exam question I tried to make up to test myself on trigonometry, but I am now confused... An example of the style of question is: The function f is defined such that $f(x) = 10 \cos x ...
2
votes
2answers
21 views

Prove that, $\left\lfloor{\frac{x}{n}}\right\rfloor=\left\lfloor{\frac{\lfloor{x}\rfloor}{n}}\right\rfloor$ where $n \in{\mathbb{N}}$ [duplicate]

Prove that $$\left\lfloor{\frac{x}{n}}\right\rfloor=\left\lfloor{\frac{\lfloor{x}\rfloor}{n}}\right\rfloor,$$ where $n \in{\mathbb{N}}.$ My Attempt: Let $x=nt$. Then, I need to prove, $$\...