Questions tagged [functions]

For elementary questions about functions, notation, properties, and operations such as function composition. Consider also using the (graphing-functions) tag.

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13 views

Bijective concave function from reals to reals

I'm looking for an unbounded function $f(x)$ such that $\forall x \in \mathbb{R}$, $$ \frac{\textrm{d} f}{\textrm{d} x} > 0\quad\textrm{and}\quad \frac{\textrm{d}^{2} f}{\textrm{d} x^{2}} < 0. $$...
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1answer
32 views

An onto map to a Unit Circle $|z|=1$

An onto map from the critical line $$\frac{1}{2}+it , t\in \mathbb{R}$$ to the unit circle $$|z|=1$$ My try- $$z= \frac{\frac{1}{2}+it}{\frac{1}{2}-it} $$ $$|z|=1$$ But the map is not onto $z=-1$ is ...
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40 views

Proof that two sets are equal

$X$ is finite, if and only if, the affirmation is true, $$Y\subseteq X \text{ and } f:Y\to X\text{ surjective } \Rightarrow Y=X$$ It is easy to see that this result is true, but I am not able to ...
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4answers
47 views

Find the function $f(x)$.

Let $f(x)$ be a polynomial function. If $f(x+2) - f(x) = 8x - 2$ and $f(0) = 5$, then what is $f(x)$? I tried to replace $x$ with $0,2,4,\ldots $ for discovering some regular pattern but I have no ...
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24 views

Applying repeated doubling to an update Matrix

Given a rule to obtain a new $x,y$ positions from an initial $x_0,y_0$ The rule is $ \binom{x'}{y'} = \begin{pmatrix} v_x & -v_y \\ v_y & v_x \end{pmatrix} \binom{x}{y}$ $x' = v_x x - v_y y$ ...
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18 views

Making a $\pi$-periodic function into a $2\pi$-periodic one

Given the function $$f(x)=\sin(x)$$ for $x\in[0,\pi]$, I have to find its even expansion for all real number such that its period is $2\pi$. What I thought was to take $$g(x)=|\sin(x)|$$ which would ...
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0answers
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Formula to determine amount threshold in T&E for audit sampling

I am in process to define amount threshold for T&E expense which would be picked for sampling but not sure what formula to be used. Please assist.
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Find all functions that satisfy $(f(x)+f(y))(f(z)+f(t))=f(xy-zt)+f(xt+yz)$ for all real $x, y, z, t$ and $f:\mathbb{R} \rightarrow \mathbb{R}$ [closed]

Find all functions that satisfy $(f(x)+f(y))(f(z)+f(t))=f(xy-zt)+f(xt+yz)$ for all real $x, y, z, t$ and $f:\mathbb{R} \rightarrow \mathbb{R}$ I got that $f(0)=0$ and $f(x)f(y)=f(xy)$.
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What is wrong in this choice of $\delta$?

Suppose I have a wrongly evaluated limit$-$ $$\lim_{x \to2} 3x+3=6$$Then we get $-$ $$3x+3-6<\varepsilon $$ $$3x-3<\varepsilon $$$$|x-1|<\frac\varepsilon {3}$$$$|x-2|<\frac\varepsilon 3 -...
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1answer
57 views

Why is $f_1(n)$ not computable but $f_2(n)$ is?

I have the following two functions, where the first one is not computable and the second one is. $$f_1(n)= \begin{cases} 1 & ,\text{if in the decimal representation of n appears in the ...
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4answers
30 views

If $\mathbb{X}$ is a set then $F^x$ denote the set of functions from $\mathbb{X}$ to $\mathbb{F}$. What does this mean?

If $\mathbb{X}$ is a set then $F^x$ denote the set of functions from $\mathbb{X}$ to $\mathbb{F}$. What does this mean? It is a definition from Sheldon Axler's Linear Algebra textbook. Please explain ...
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2answers
32 views

Continuum between arithmetic mean, AGM, and geometric mean

NOTE: I am aware of this possible duplicate, but my question is slightly different as it also involves arithmetic-geometric mean. The arithmetic mean of two numbers is defined as: $$\text{am}(a,b) = \...
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22 views

How many stationary points does $y=f\left(g(x)\right)$ have for $1 \le x \le 5$?

The graph shows two functions $y=f(x)$ and $y=g(x)$. Define $h(x)=f\left(g(x)\right)$. How many stationary points does $y=h(x)$ have for $1\le x \le 5$? The correct answer is 3, but I do not see why. ...
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2answers
35 views

The Question about Euler's Totient Function : How phi(i*p) = p*phi(i)? Here, p is prime and p divides i, phi-> Euler totient function

I have a question about the Euler totient function. I am new to the number theory and i don't know where to start to prove this. If $p$ is prime and $p$ divides $i$, $$ Φ (i\cdot p) = p \cdot Φ (i),$$...
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1answer
44 views

If $f(0)=0,f(x)<x,x_{n+1}=f(x_n)$ and $f'(0)\in(0,1)$, then $\sum x_n$ converges. [closed]

Let $f:[0,1] \to [0,1]$ a class $C^1$ function satisfying $f(0)=0$ and $f(x)< x,\forall x \in (0,1)$. Define a sequence $x_n$ recursively by choosing $x_1 \in (0,1)$ and putting $x_{n+1}=f(x_n)\,,\...
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2answers
47 views

Prove that $𝑓$ is injective and $𝑔$ is surjective [closed]

This is the question I got and I'm not sure how to approach it. Let $f: A → B$ and $g: B → A$ be two functions, such that $g ∘ f(x) = x$ for all $a ∈ A$. Prove that $f$ is injective and $g$ is ...
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Conjecture on simple algebraic expressions for complemetary sets

This question/conjecture started in a different discussion regarding arithmetic series. Conjecture: If we have a set that is defined by a single algebraic expression, e.g., $$A := \{a \in \mathbb{Z}^+ ...
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1answer
34 views

A polynomial function that is non-constant is injective iff it contains one root that is non-zero, or no roots at all.

I was messing around the other day in desmos, and I began to graph functions which have certain roots, (for instance $x^2-x-1$), and noticed that these functions are not injective. In this case, $\phi$...
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1answer
31 views

Function connected implies Darboux function

Definition Let $f: X \rightarrow Y$ a function, $X,Y$ topological spaces. We say that $f$ is connected if $G_{\text{f}}(f)= \{(x, f(x)): x \in X\}$ is connected and we say $f$ is Darboux if $ \forall ...
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There exist odd functions $g : R → R$ and $h : R → R$ such that $f(x) = g(x) − h(x)$ is a non-constant even function.

I tried solving on but I am stuck on what is non-constant even function and how it is related to this question
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1answer
49 views

What is the record for Collatz Conjecture Steps

Does anyone know the record for the number of steps a Collatz Conjecture run has taken to get to 1? I have written a small program in Python to do the Collatz Conjecture on the Mersenne prime where n=...
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2answers
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An explicit polynomial isomorphism between the real projective line and the unit circle.

Prove that the map $$w=\frac{\frac{1}{2}+it}{\frac{-1}{2}+it} , t\in \mathbb{R}$$ is a bijective map which maps the real axis $(-\infty,\infty)$ to the unit circle $|w|=1$ My try $$w=\frac{\frac{1}{2}+...
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29 views

How to prove the function $f(x)=260-x+\sqrt{x^2+3600}$ is injective? [closed]

I have to prove the injectivity of this function but i don't know how to solve this. I tried using the definition, but i am having difficulty figuring out what to do.
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1answer
56 views

Proving a general result about an iterated function

I recently found a question that stated as follows: Let $f:\mathbb N\to\mathbb N$ be defined as follows: $$f(n)=2n+1-2^{\lfloor\log_2n\rfloor+1}$$ and let $f^\alpha(n)=f(f^{\alpha-1}(n))$. Let $t(n)$ ...
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1answer
31 views

How many function $f\colon A\to B$ with the following properties: $f (x + y) = f (x) + f (y)$ for every $x, y\in A$

Given $A = \{1, 2, 3,\dots , 111\}$ and $B = \{1, 2, 3,\dots , 2021\}$. How many function $f\colon A\to B$ with the following properties: $$f (x + y) = f (x) + f (y)$$ for every $x, y \in A$. All I ...
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36 views

Show that $ f $ is differentiable and that $ f ': \Omega \rightarrow L(\mathbb{R}^m; \mathbb{R}^n) $ is continuous.

Let $\Omega=\{ T \in L(\mathbb{R}^m;\mathbb{R}^n) : \mbox{T is invertible}\}$. Consider $f: \Omega \rightarrow \Omega $ given by $ f(A) = A^{-1} $. Show that $ f $ is differentiable and that $ f ': \...
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2answers
72 views

Two sets have the same cardinal proof

Suppose that $S, T$ are sets such that $S$ is infinite and $T = \{t_i: i = 1, ..., n\}$ with $T$ having $n \geq 1$ distinct elements. If $S \cap T = \{\}$, show that $card(S) = card(S \cup T)$. Hint: ...
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1answer
18 views

Functions and natural ordering

Imagine that we have the following function: $$ Y=f(X) $$ where you can imagine $Y$ to be anything- say quantity of food consumed, and $X$ is income. Now, when we think about counting objects, an ...
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0answers
31 views

Show that all functions $f_k(x)=\sin(kx)$, $k\in\mathbb{R}$ have at least one $x$ where all $f(x)_k\geq\frac{1}{7}$ [closed]

In connection with waves in physiks I have come up with the following question: Is there some $x$ for which all $n$ functions of the form $f_k(x)=\sin(kx)$ with the parameter $k\in\mathbb{R}$ are ...
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41 views

Number of steps to find the number of primes less than or equal to $x$.

Denote by $\pi(x)$ the number of primes less than or equal to $x$ (the prime counting function). I am curious how many steps it takes to find $\pi(x)$, where a step is defined is an elementary ...
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14 views

A solution for a general functional equation.

I recently found a problem that stated Show that there does not exist a function $f(n)$ so that $f(f(n))=b+1987$. I solved this problem, but doing so, I made this claim and I wish to see if it is true....
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2answers
30 views

Let $f_n(x)=\frac{1}{nx+1}$ and $g_n(x)=\frac{x}{nx+1}$ for $x\in (0,1)$. Show that $f_n$ doesn't converge uniformly but that $g_n$ does

Let $f_n(x)=\frac{1}{nx+1}$ and $g_n(x)=\frac{x}{nx+1}$ for $x\in (0,1)$. Show that $f_n$ doesn't converge uniformly on $(0,1)$ but that $g_n$ does. I know that it is not something difficult, but I ...
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1answer
28 views

Why is the derivative of f equal to the sum of its partials along its components?

I was trying to understand the development of the solution in this answer, where $\overline{f(z)}f'(z)dz = (u - iv)(\frac{\partial u}{\partial x} + i \frac{\partial v}{\partial y})(dx + idy)$. The ...
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1answer
79 views

Proving that $f$ verifies $f(1−x) + f(x) = 1$

I have a function which is defined for $x \in (0,1)$ and for $p>1$ with the expression \begin{align*} f(x) = \sum_{k=0}^{p-1} \frac{(-1)^{p+k}}{(p-1-k)!(p+k)!}\left(\prod_{i=k-p+1, i\neq0}^{k+p} (...
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Functions similar to $f(x) = x^2 \sin{\frac{1}{x}}$

I am analysing this function: $f(x) = x^2 \sin{\frac{1}{x}}$ The specific feature of this function that I am interested in is the increasing smoothness as you move away from zero. Are there similar ...
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2answers
33 views

Intersection of a exponential and a sin wave

I need to find the intersection between to functions to alter the domain so that they connect at said intersection. $$0.01^{x-17.7}+7 = \sin\left(\frac {x}{1.3} + 2.6\right) + 8$$ This is for a year ...
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0answers
25 views

When is $g(t):=\inf_{s \in (0,t)} f(s)$ differentiable?

Under which non-trivial conditions is the function $$g(t):=\inf_{s \in (0,t)} f(s)$$ differentiable? Would the differentiability of $f\colon \mathbb{R} \to \mathbb{R}$ be enough? Where can I find some ...
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0answers
20 views

Contractive Fixed Point Theorm for Vector Value Functions

Consider the following question: I tried to look on the function $g(x)=f(x)+[x_1,x_2]$ , so if I could prove that $g(x)\in S$ for $x\in S$, then by the contractive fixed point theorm there exists ...
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21 views

prove this question regarding set theory and functions [closed]

[please help me with this][1] The question is in relation with set functions namely onto . [1]: https://i.stack.imgur.com/NMWYN.jpg
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How to determine the pre-image point for this transformed function?

A function $f(x)$ is transformed to $2f(-7x-4)+10$. Let $(6,5)$ be a point on the transformed graph. What is $x$-coordinate of the corresponding pre-image point?
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22 views

Finding a distribution name by its formula

I'm looking for a search engine which given the formula of function returns the name of the associated function or probability distribution, if any, up to constants. For instance, if I input: $$ f = k ...
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1answer
64 views

Is it true that the domain of this function is finite?

(This is a subproblem that came up as I was proving that $\{e: \phi_e(x) \text{ has infinite domain}\}$ is $\Pi_2$-complete.) Let $A$ be a set of natural numbers and suppose $$n\in A\iff \forall y_1\...
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2answers
40 views

How to express a formula that iterates over itself n number of times?

I'm not very good at math so I might use the wrong words when searching for an answer (i even don't know what tags to apply). I have an expression in a loop and the only way I can explain it is ...
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2answers
38 views

Question: For what value of $c$ will $P(x)= -2x^3+cx^2-5x+2$ have the same remainder when devided by $x+1$ or $x-2$?

For what value of $c$ will $P(x)= -2x^3+cx^2-5x+2$ have the same remainder when devided by $x+1$ or $x-2$? No idea where to start with this one. Can't use the remainder theorem to find the remainder ...
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2answers
39 views

Average of a pair with respect to a function

I have $200$ units of good A and $200$ units of good B. For each unit, I assign them an integer price $[1, 2, 3, ... 200]$. Each price can only be assigned to one unit of good A and B. For example, ...
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2answers
49 views

Show that function composition is not commutative. Find a suitable $f(x)$ and $g(x)$ such that $(f \circ g)(x)\neq(g∘f)(x)$ [closed]

In this context (in working with a function under the operation of composition) when we raise a function to a power like $f^2$, this means $(f \circ f)(x)$. In other words, we apply the composition ...
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0answers
30 views

How to find number of “Multiplying Functions”?

Consider the set $A=\{1,2,3,4,5\}$. We call the function $f$ "Multiplying" if for each $x,y\in A$, at least one of these two conditions satisfied: $x\times y>5$ $f(x\times y)=f(x)\times ...
6
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1answer
172 views

Proving that $f$ verifies $f(1-x)=-f(x)$

I'm trying to show an identity verified by a function. I have this function which is defined for all real numbers as $ f(x) = \sum_{k=0}^{p-1} \frac{(x+k)^{2m-1}}{(-1)^{p+k} (p-1-k)!(p+k)!}$ I would ...
0
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1answer
32 views

Number of functions from $A$ to $A$ [duplicate]

We have the set $A=\{1,2,3,4,5\}$. what is the number of functions from $A$ to $A$? I think the domain of the function should contain all the elements of $A$ but range of the function may have any ...
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1answer
48 views

Can a function from an interval to a set of rational numbers be bijective?

I have the next function: $$g:[0,1]\rightarrow(0,1)\cap\mathbb{Q}$$ Can it be bijective? Or is it at most surjective or one-to-one? I believe that because there are more elements in the domain than ...

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