# 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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### 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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### 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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### 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 ...
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### 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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### 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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### 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 ...
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### 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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### 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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### 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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### 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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### 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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### 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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### 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 ...
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### 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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### 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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### 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 ...
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### 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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### 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 ...
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### 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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### 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 ...
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