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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Periodic function with period 2

Let f(x) be a periodic function of x with period 2 and f(x) = |x| − x for −1 < x ≤ 1. Sketch the graph of the curve y = f(x) in the interval [−3, 3]
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1answer
16 views

Show g is not surjective

$\begin{align} f: A \rightarrow B, \space g : B \rightarrow A, \end{align} $ and $\begin{align} f \circ g : B \rightarrow B \end{align}$ is bijective I am not sure how to show that $g$ doesnt have to ...
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0answers
17 views

A function f is defined by , find the implicit domain of f (please) [closed]

A function f is defined by enter image description here find the (implicit) domain of f
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1answer
356 views

Expected number of cycles in random function

Let f be a left-total function with a discrete domain and codomain both in the range of 0..N. The image of f is chosen randomly (with replacement). Given N, how many cycles is the function expected to ...
1
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2answers
946 views

Prove that $\sqrt{x}{\sin \frac1x}$ is uniformly continuous at $(0, \infty)$

I need to prove that $f(x) =\sqrt{x}{\sin \frac1x}$ is uniformly continuous at $(0, \infty )$. I managed to show that: $-\sqrt{x} <= f(x) <= \sqrt{x}$, $\lim_{x\to0}\sqrt{x} = \lim_{x\to0}-\...
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0answers
40 views

The existence of a strange function on $[0,1]^2$ [duplicate]

Does such a function $f:[0,1]^2\rightarrow \mathbb{R}$ exist: $f(\cdot,y)$ and $f(x,\cdot)$ are continuous functions with respect to "$\cdot$", for any $x,y\in[0,1]$; The zero set of $f$ ...
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0answers
30 views

Multinomial distribution as a weighted distribution?

Let $f_1$ be the following multinomial distribution: $$f_1((k_1,...,k_m);n,(p_1,....,p_m)) = \frac{n!}{\prod_i k_i!}\prod_i p_i^{k_i} $$ where $\sum_i k_i = n$ and $\sum_i p_i = 1$ Let $f_2$ be the ...
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0answers
37 views

$f(x)=\lim\limits_{t\rightarrow0}\frac{\sin(\sin(k\pi/e^{1/t})e^{1/t}[x^2-x+\pi])}{\ln (k+[x]^2)}$ where k is an integer and $x\in \mathbb R$

$f(x)=\displaystyle\lim\limits_{t\rightarrow0}\frac{\sin(\sin(k\pi/e^{1/t})e^{1/t}[x^2-x+\pi])}{\ln > (k+[x]^2)}$ where [.] denotes the greatest integer function, and k is an integer. For what ...
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3answers
165 views

How can I solve the functional equation $ f ( x + 1 ) + 1 = f \big( f ( x ) + 1 \big) $?

Let $ \mathbb N = \{ 0 , 1 , 2 , \dots \} $. Find all the functions $ f : \mathbb N \to \mathbb N $ such that $$ f ( x + 1 ) + 1 = f \big( f ( x ) + 1 \big) $$ for all $ x \in \mathbb N $. I noticed, ...
20
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1answer
700 views

Find a function such that $f^{-1}=f'$

Let $f:\Bbb{R}^+\rightarrow\Bbb{R}^+$ be a differentiable bijection and let $f$ satisfy: $f'=f^{-1}$ (where $f^{-1}$ denotes the inverse of $f$). Find $f$. This comes from a facebook page "...
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0answers
42 views

Infinite sets: Equipotent sets

I need to prove the next theorem about equipotents sets. If $A$ $ \approx$ $B$ and $C$ $\approx$ $D$ and suppose that $A$ $\cap$ $C$ $=$ $\emptyset$ and $B$ $\cap$ $D$ $=$ $\emptyset$ then $A$ $\cup$ $...
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0answers
46 views

Is the codomain the inverse image of the domain?

Starting from the definition of the inverse image, I'm wondering if, as a special case, is possible to say that the codomain of a function is the inverse image of the domain. In the classical ...
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1answer
20 views

Monomorphism if and only if injective

The definition of monomorphism in Aluffi's book is: A function $f: A \to B$ is a monomorphism if for all sets $Z$ and all functions $\alpha' \alpha'' : Z \to A$, $f \circ \alpha' = f \circ \alpha'' \...
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0answers
41 views

What is the degree of a non-polynomial function, if it has one?

For polynomials $P$ of the form $$P(x) = \sum_{i=0}^n a_ix^i,$$ the degree of $P$ is $n$. My question is this: Do all non-polynomial functions have a degree, and if it has one, what is it? From what I ...
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1answer
31 views

Are there maximum and minimum values for the exponential function $y = 100 \times {0.85^t}$ , for 0 ≤ t ≤ 6. Explain.

I don't really understand this question, if anyone could help with it it would be greatly appreciated. Want to know if $y = 100\times{0.85^t} $ where $t ∈ [0, 6]$ Has any maxima or minima?
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3answers
128 views

Let $p(x)$ and $q(x)$ be two polynomials such that $p(2)=5$, $p(3)=12$ and $p(q(x))=p(x)q(x)-p(x)$. Find the value of $q(10)$.

Let $p(x)$ and $q(x)$ be two polynomials such that $p(2)=5$, $p(3)=12$ and $p(q(x))=p(x)q(x)-p(x)$. Find the value of $q(10)$. The question is from a local mock contest that took place a few days ago....
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0answers
20 views

finding sec inverse range

Let a function Domain D and Range([0,pi]-A) , A →{b1 , b2 , b3 , ...} $$f(x) = \sec^{-1}\biggl( \frac{2x^2 -x -1}{1 - x^2} \biggr)$$ Find number of $n(A)$ so that the function $f$ is onto.
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2answers
953 views

How is a morphism different from a function

How is a morphism (from category theory) different from a function? Intuitive explanation + maths would be great
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0answers
31 views

Is a morphism a special case of a function?

https://en.wikipedia.org/wiki/Morphism states that In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same ...
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0answers
42 views

Constructing a function by recursion

We have the following recursion principle on $\mathbb{N}$ Let $A$ be a non-empty set and $a_0 \in A$. Let $h : \mathbb{N} \times A \to A$ be a function. Then there is a unique function $f : \mathbb{N}...
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0answers
23 views

Name of function that maps to input set

Let $f: S \rightarrow S$ be a function and $S$ be a set. $S$ is not the underlying set of a vector space. Most likely, it is also not the underlying set of a topology (though I am not sure). It would ...
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4answers
511 views

If $(f\circ g)(x)=x$ does $(g\circ f)(x)=x$?

Given $$(f◦g)(x)=x$$ (from R to R for any x in R) does it mean that also $$(g◦f)(x)=x$$ I feel like its not true but I can't find counter example :( I tried numerous ways for several hours but I ...
2
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4answers
125 views

nth time differentiability at a point if limit of nth derivative exists at that point

Exercise: Let the function $f$ be defined and continuous in an open interval $A$. Suppose that $c$ is a point in $A$ and that $f$ has derivatives up to order $m$ on the set $A \backslash\{c\}$. ...
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1answer
41 views

Composite functions equal to identity problem

Find example of functions f,g : N → N (N= {0, 1, 2, . . . }), so that f ◦ g = idN and simultaneously g ◦ f ≠ idN (or justify, why those functions don't exist.) I'm having problem solving this one. I ...
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0answers
29 views

Lambda notation for abstraction (Set theory).

I'm trying to solve a problem in Suppes' "Axiomatic set theory". It's about functions and lambda notation for abstraction, in the framework of ZFS set theory. I've defined, as usual, ...
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0answers
16 views

What is the difference between deconvolution and curve fitting? [closed]

I usually see both terms used interchangeably but I think they represent two different terms although related.
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1answer
55 views

Prove that the Collatz function is not surjective no matter what initial condition we choose.

Where the initial condition is $f(0)=k$ , where $k \in \mathbb{N} $ So I just started reading some discrete mathematics and this problem came up in Oscar Levin's book " An open introduction to ...
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0answers
21 views

How to bind conditions on a Quadratic/Quadratic function [duplicate]

For example,we need to find the range of a for which expression $\dfrac{ax^2+3x-4}{3x-4x^2+a}$ assumes all real values for real values of $x$ how to proceed?
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2answers
28 views

If $f(x)$ is a concave upward function in the interval $x_1,x_2$, prove that $f\left(\frac{a+b}{2}\right)\leq \frac{f(a)+f(b)}{2},x_1\leq a,b\leq x_2$

If $f(x)$ is a concave upward function in the interval $x_1,x_2$, prove that $f\left(\frac{a+b}{2}\right)\leq \frac{f(a)+f(b)}{2},x_1\leq a,b\leq x_2$ The slope of a concave upward function increase ...
0
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1answer
90 views

Is there increasing function on $(0, \infty)$ such that $\lim_{x\to 0^+} \frac{f(x)}{x}=\infty$? [closed]

Let $f:(0,\infty)\mapsto (0,\infty)$ be a non-decreasing function such that $f(x)\to 0$ iff $x\to 0$. Is it possible that $$ \lim_{x\to 0^+} \frac{f(x)}{x^k}=+\infty$$ for each $0<k<1$? I feel ...
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1answer
37 views

If $\ f\biggl(\frac{2 \tan x}{1+\tan^2x}\biggl)=\cos (2x+1)\cdot(\frac{\sec^2x+2\tan x}{2})$ then domain of $f(x)$ is?

If $\ f\biggl(\frac{2 \tan x}{1+\tan^2x}\biggl)=\cos (2x+1)\cdot(\frac{\sec^2x+2\tan x}{2})$ then domain of $f(x)$ is? Teacher's Statement: Domain is [-1,1], because $\frac{2\tan x}{1+\tan^2x}=\sin(...
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3answers
86 views

floor function equation (⌊x⌋=⌊x^2⌋)

ex : solve in R ⌊x⌋=⌊x^2⌋ (or floor(x)=floor(x^2) my answer sorry its in french and its poorly written
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2answers
25 views

Finding range of a log function with variable base

How can I find the range of $\log_{x-7}(x-5)$? Here is my attempt. Let $$y=\log_{x-7}(x-5)$$ Let $z=x-7$, $y=\log_z(z+2)$ then $y$ satisfies $$z^y-z-2=0, z>0, z \ne 1.$$ $y$ be written as $\frac{\...
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1answer
66 views

Is there a specific name for $Y=(X-c)^+$, as a function of a random variable?

Let $X$ be a random variable, and for a given $c>0$ let $f_c:\mathbb R\to [0,\infty)$ be a measurable function defined by $x\mapsto \max\{x-c,0\}$. Write $Y:=f_c$. I have not been able to find a ...
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0answers
25 views

Using logical operators in the domain constraints of functions

Is it 'mathematically correct' or 'acceptable' to use logical operators when defining the domain of a function? Say, I have $$f(x,y)=\frac{\sqrt{x^2+y-1}}{x^2-1},$$ then for its domain, I will have $$\...
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1answer
19 views

Interval "Widths"

If $h(x)$ is a function whose domain is $[-8,8]$, and $g(x)=h\left(\frac x2\right)$, then the domain of $g(x)$ is an interval of what width? Hello, I recently found myself stuck with this problem. ...
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1answer
82 views

A doubt about the fundamental concept of the calculus about continuity.

I'm confused with a fundamental concept of calculus about continuity. I'm studying the concept of continuity and discontinuity of a function. The case is the following: "It makes no sense to talk ...
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1answer
18 views

Mathematical expression of cumulative sum reset function

Could anybody please help me to formulate an expression for the function that sums up observations along a vector, but wherever the sum becomes negative it reset the the sum to 0 and starts suming up ...
0
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1answer
1k views

What type of function is this (derivative of a hyperbola)?

The derivative of the hyperbola $$f(x)=\frac{b}{a}\sqrt {a^2+x^2}$$ is $$f'(x)=\frac{bx}{a\sqrt {a^2+x^2}}$$ The graph (for $a=b=1$) looks somewhat like a Sigmoid function, but I honestly cannot ...
0
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1answer
36 views

Degree of the homogenous function $f(x)=0$

A homogenous function is a function such that, when its arguments are multiplied by a scalar, the value of the function is multiplied by some constant power of that scalar, e.g: $$f(kx) = k^nf(x), \...
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0answers
13 views

Lagrangian method when the constraint is a triangle [closed]

Can you use the Lagrangian method to find the maximum and the minimum value of the function: $f(x,y)=xy-x-y+3$ in the triangle with the vertices at $(0,0), (2,0), (0,4)$. If you can, how would you ...
0
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2answers
52 views

Square-root function

Recently, I was learning about square-root function which has domain [0,∞) and range is [0,∞). I understand that domain has to be positive numbers but why range is restricted to positive numbers?
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0answers
60 views

Amazing result that has no proof anywhere [duplicate]

This is the identity that I came up with when I was solving the Frobenius number problem $$\sum_{j=1}^n \hspace{0.2cm}[\frac{j\cdot n}{m}] = 0.5(n-1)(m-1)$$ Where [x] means floor(x) = greatest ...
-1
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0answers
61 views

any function can be presented as a composition of surjective and injective functions [closed]

Prove that arbitrary function $𝑓:A\to B$ can be written as a composition between a surjective and an injective function, respectively ($f = g(v(x))$, where $v$ is injective and $g$ is surjective).
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1answer
53 views

Function $f$ on plane satisfying $f(I)=f(A)f(B)f(C)$ for any triangle $ABC$ with incenter $I$

Let $f$ be a function from the set of all points on the plane to the nonzero real numbers. Suppose that for any triangle $ABC$ with incenter $I$, we have that $f(I)=f(A)f(B)f(C)$. What are the ...
1
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1answer
36 views

Can a general function that flips the two arguments of Atan2 be found?

What is the general form of function $F$ if $F(\operatorname{atan2}(y,x))=\operatorname{atan2}(x,y)$? In words: I seek a function that flips the order of the two arguments. By plotting $\operatorname{...
0
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0answers
38 views

How can I demonstrate $f(g(x))=h(x)$ implies $f(x)=h(g^{-1}(x))$ taking all it to be bijective

I'd actually like to demonstrate it not proof that is using algebra to get $f(x)=h(g^{-1}(x))$ and not verifying $f(g(x))=h(x)$ by $f(x)=h(g^{-1}(x))$ I've tried: $f.g=h$ $(f.g).g^{-1}=h.g^{-1}$ $f=h....
0
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1answer
48 views

What function could map all negative integers to between 0 and 1 and keep all positive integers the same?

I have an input that can be any integer between -infinity and +infinity. I would like to find a function that maps all positive integers to themselves (1->1, 2->2 etc) but maps all negative ...
3
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0answers
17 views

Proving the existence of a condition for reciprocal of derivatives of a given function

Let f be any continuously differentiable function on [a,b] and twice differentiable on (a,b) such that f(a)=a and f(b)=b. Then prove that there exists some distinct $c,d$ such that $$1/f'(c)+1/f'(d)=2$...
2
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0answers
50 views

Inverse functions and $f(x)=x$

Is every solution of $f(x)=x$ a solution of $f^{-1}(x) = f(x)$? If not, why not? Can we not do the following ? \begin{eqnarray} f(x)&=&x\\ f(f(x))&=&f(x)=x\\ f^{-1}(x)&=&f(x)\...

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