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

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3
votes
1answer
17 views

What is the difference between functions and operations?

Wikipedia says that an operation $\omega$ is a function of the form $\omega: V \to Y$, where $V \subset X_1 \times\cdots\times X_k$. But as far as I know, every function's domain is a set, so ...
1
vote
1answer
20 views

Onto (surjective) functions of 2 variables [closed]

I have a couple of functions I'm curious about: $f(m,n)=m^2 -n^2$ and $f(m,n)=|m|-|n| $, for $m,n\in \mathbb{Z} $. The codomain also consists of all integers. My understanding is that for this ...
1
vote
1answer
15 views

Is it possible to express the indicator function for a real interval in terms of other function/s?

Suppose that I've an indicator function defined for an interval in $\mathbb R$, i.e., suppose that $f(x)=1$ if $x\in(a,b)$ and $f(x)=0$ otherwise. Then can I express $f$ without using indicator ...
5
votes
3answers
4k views

Relation between differentiable,continuous and integrable functions.

I have been doing lots of calculus these days and i want to confirm with you guys my understanding of an important concept of calculus. Basically, in the initial phase,students assume that ...
0
votes
1answer
27 views

What would the multifunctional inverse of $F(x)=|x|$ be?

What would the multifunctional inverse of $F(x)=|x|$ be, assuming $x$ is on the complex plane. Also, how would this usually be represented? Note that this won't be a 'true' function. (But assume a ...
0
votes
0answers
14 views

Extension of a quasiconvex function

A function $f$ defined on a subset $S$ of the $n$ dimensional Euclidean space is said to be quasiconvex if $f(ax + (1-a)y) \le\max \{f(x) , f(y)\}$ for all $x, y \in S, a \in [0,1]$. Now, suppose ...
1
vote
2answers
73 views

Is 'clamp' a formally recognized mathematical function?

I was surprised to find the clamp function absent from Mathworld and Wikipedia. Is this mainly a concept particular to computer programming? Is it known by another ...
0
votes
2answers
29 views

Is the composing of functions always commutative?

I have a question for my math study. It seems quite simple, but I just can't find a counterexample for the following: The composition of two functions is always commutative Could you help me with ...
0
votes
1answer
33 views

Why is the following true? functions

$$x , x_0 \in [a,b]$$ $x_0$-fixed $f \in D(a,b)$- differentiable on [a,b] $$\triangle (x)=f(x)-f(x_0)-f'(x_0)(x-x_0)$$ $$\triangle '(x)=f'(x)-f'(x_0)$$
2
votes
1answer
9 views

Finding the function f(t) from it's graph

Here's what I have so far: $$f(t) = (-2(t+1)+1.5) \times (u(t+1)-u(t)) + (t-0.5) \times (u(t)-u(t-1)) + 0.5\cos(\pi t) \times (u(t-1)-u(t-3))$$ I found the majority of this function, but I'm not ...
4
votes
0answers
75 views

What function satisfy: $f(x)+f^{-1}(x)=2x$?

What function satisfy: $f(x)+f^{-1}(x)=2x$? I have tried to substitute $x=f(x)$ to get $f^{(2)}(x)+1=2f(x)$ and subsequently plug in values to try to find $f(x)$ but to no avail. Please help thank ...
0
votes
1answer
60 views

Inverse function $g^{-1}$

The function $g$ is defined by $$g(x)= 3-2x-4x^2, x\in \mathbb{R},x\leq -\frac{1}{4} $$ Find the inverse function $g^{-1}$. Calculate the value of $x$ for which $g(x)=g^{-1}x$. My attempt, ...
1
vote
0answers
44 views

Given x,y,w,h can you generate a rainbow box/cuboid with rounded edges?

Given $x$, $y$, $w$, $h$ where $0 \leq x < w$ and $0 \leq y < h$ and $(x, y)=(0, 0)$ is bottom-left and $(x, y)=(w-1, h-1)$ is top-right and they're all integers, can you make a formula that ...
0
votes
3answers
48 views

How can I check if a function is always $\ge 0$ for all values of its parameter $\ge 0$?

Let's say I have a function with with arbitrary coefficients and powers, something like $f(x) = 5x^2 - 7x^3 + x^4$. How can I check if this function is always $\ge 0\ \forall x \ge 0$? The procedure ...
0
votes
4answers
64 views

If neither of $f: A \to B$ or $g:B \to C$ is one-to-one can $ g \circ f$ be one-to-one?

The title says it all - but to reiterate: If neither of $f: A \to B$ or $g:B \to C$ is one-to-one can $ g \circ f$ be one-to-one? I think not. Anyone have a good proof for this? This is simply ...
2
votes
3answers
24 views

Function and Domain with Deleted Neighbour - Beginner Question

I have a simple question about functions and domains. Consider the following function: $$f(x) = \frac{ x^2-9}{x-3}$$ I often see in the textbooks mentioning that the domain of this function can be ...
0
votes
1answer
6 views

Convex combination of quasiconvex functions.

Is a convex combination of two quasiconvex functions necessarily quasiconvex? If not, what can be said about the convex combination?
-2
votes
2answers
65 views

Prove $f(A_1 \cap A_2 \cap A_3 \cap \dotsb \cap A_n)=f(A_1) \cap f(A_2) \cap f(A_3) \cap \dotsb \cap f(A_n)$ [closed]

Let $f: R \to R$ be a one to one function. For any collection of subsets $A_1, A_2, A_3, \dotsc A_n$ of $R$, prove that $$f(A_1 \cap A_2 \cap A_3 \cap \dotsb \cap A_n)=f(A_1) \cap f(A_2) \cap f(A_3) ...
9
votes
3answers
233 views

Quadratic Equation Recurrence?

So I was playing around with the following: Let $f_0(x) = x^2 - bx + c$. If $f_n(x)$ has roots $p$ and $q$ with $p > q$, then let $f_{n+1}(x) = x^2 - px + q$. The recurrence relation is rather ...
1
vote
2answers
20 views

Product of two continuous, non-negative and monotone non-decreasing function is itself..

My question is simple: is the product of two continuous, non-negative and monotone non-decreasing function itself a continuous, non-negative and monotone non-decreasing function? I believe the ...
4
votes
1answer
99 views

Multitangent to a polynomial function

I'm trying to build some exercises on tangents of functions for beginner students in mathematical analysis. In particular I would like to suggest the study of polynomial functions $ y = p (x) $ of ...
1
vote
1answer
13 views

Showing the following sequence is monotone decreasing

Let $T$ be fixed and define the functions $$a_k(t) = \frac{e^{\mu_k (T-t)} - e^{-\mu_k(T-t)}}{e^{\mu_k T}- e^{-\mu_k T}}$$ for $t \in [0,T]$. Given that $\mu_k$ is a monotonically increasing ...
2
votes
1answer
35 views

Function that Represents Divergent Power Series?

Suppose we have the following power series $$\sum_{k=0}^\infty\left(x^2+1\right)^{2k}$$ If we wished to find the function that represents this series, it seems reasonable to suppose that the ...
1
vote
2answers
905 views

Trying to figure out a formula with given input and outputs.

I'm playing this video game where people can get kills, deaths, and assists , and all this is recorded on a stats website. The stats website gives you a rating by directly manipulating these numbers. ...
0
votes
2answers
18 views

Find all critical numbers of $f(\theta) =\cos(2\theta), 0 < \theta < \frac{\pi}{4}$ then use first derivative test to find rel max or min

The theta and cosine are really throwing me off. I had one of my friends help me out, but the work really does not make sense to me. They got $\sin(0) = 0$ is the only critical number and there is no ...
3
votes
1answer
28 views

Degree Of Polynomial Factored Function $g(x) = 0.5x (x+4)^2(2x-3)$

I'm confused about the process of how to find the degree of a polynomial factored function. I'm not sure that in this specific question if there is only two zeros/factors or if the 0.5x also plays ...
5
votes
5answers
249 views

Theoretical function question

Suppose we have the function $f(x)= x^2 $. This function associates real numbers with real numbers ( $f:\mathbb{R}\rightarrow \mathbb{R}$). Now, what i get confused sometimes is what exactly the ...
1
vote
1answer
13 views

Implement a y-ceiling on a slope function without domains

I'm working on a computer software application, dealing with very large quantities of numeric data, and I'd prefer to remove the domain conditions from this graph to help boost performance. Right now ...
1
vote
2answers
35 views

If $f\colon X\to Y$ is injective there is a $g\colon Y\to X$ such that $g\circ f=Id_X$

How to prove that if $f\colon X\to Y$ is injective there is a $g\colon Y\to X$ such that $g\circ f=\operatorname{id}_X$. I know that it is an if and only if, but I have already proved the reciprocal. ...
0
votes
2answers
19 views

Understanding functions of matrices

Given $$f(X) = rank(X) $$ with X being a matrix. Is it possible to visualize such a function? What is the space that it lives in (assuming all entries live in $\mathbb{R}$)? Is there an literature ...
0
votes
1answer
15 views

How to prove that a function is convex.

Let $h:[0,1] \to (0,\infty)$ be a continuously differentiable function such that the following inequality is true: $$\frac{h'(t)}{h(t)} > -\frac{1}{2} \ \ \ \ \text{for all $t\in (0,1)$} .$$ Let ...
2
votes
0answers
47 views

Fixed Points of Function from Rationals to Reals

Consider a function $f$ from the positive rationals to the reals such that $f(x)f(y)\ge f(xy)$ and $f(x+y)\ge f(x)+f(y)$. Further assume this function has a fixed point greater than $1$. Prove this ...
2
votes
1answer
27 views

Show $\phi(x) = (x - x_1)(x - x_2) \cdots (x - x_m)$ is odd for $m$ odd.

Given the function $$ \phi(x) = (x - x_1) (x - x_2) \cdots (x - x_m) $$ where $m$ is odd, and the points $x_1, x_2, \cdots, x_m$ are symmetric wrt the midpoint of its domain, show that the function ...
2
votes
1answer
22 views

Suppose $f\leq g,h$. If $\{ f<g\} \cap \{ f < h \} = \emptyset$, then $\{f<g \} \cap \overline{\{f < h\}} = \emptyset$.

Suppose $X$ is a Hausdorff topological space. Assume that $f,g:X \rightarrow \mathbb{R}$ are continuous functions such that $f \leq g$. Denote $\{f< g \}=\{ x \in X: f(x)<g(x) \}$. Lemma: ...
0
votes
1answer
31 views

Proving that a function is increasing

I have this problem Suppose that $f: \mathbb{R} \to \mathbb{R}$ is a function, such that its Taylor series convergers to function $f$ everywhere. For every derivative of the function $f$ we have that ...
2
votes
1answer
46 views

Question on construction of entire functions

Suppose that $x_i$ and $y_i$ are sequences in $\mathbb{C}$. Can you construct a non constant entire function such that $f(x_i)=y_i$? What happens if $x_i$ have an accumulation point? or what happens ...
0
votes
0answers
20 views

continuity of linear functional on family of functions

If $A$ : $C[a,b]\rightarrow \mathbb{R}$ is a continuous linear functional, then $ t\mapsto A(f_{t})$ is a continuous function on $\mathbb{R}$. where \begin{align} f_t(x)= \left\{ \begin{array}{lr} ...
0
votes
2answers
55 views

Derivative of function $f(x) = \sqrt{2x}+ \sqrt{2/x}$

The derivative of function $$f(x) = \sqrt{2x}+ \sqrt{2/x}$$ Here's what I did, $$f(x) = \sqrt{2x}+ \sqrt{2/x} \\ = (2x)^{1\over2} + ({2\over x})^{1 \over 2}\\\\$$ $$f'(x)={1\over 2}(2x)^{-{1\over ...
2
votes
2answers
55 views

A difficult problem on functions

I've been trying to solve the following problem but can't wrap my head around it. Let $x$, $f(x)$, $a$, $b$ be positive integers. Furthermore, if $a > b$, then $f(a) > f(b)$. Now, if $f(f(x)) = ...
0
votes
1answer
22 views

A continuity result

Suppose (i) $f:R^n_+\to R$ and (ii) $f(x)=f(\alpha x)$, for all $\alpha>0$ and (iii) For any $x,y\in R^n_+$, if $x_n\to x^*$, $y_n\to x^{*}$, we have $\lim f(x_n)=\lim f(y_n)$. Could I claim ...
2
votes
0answers
20 views

Example of a continuous non-lipschitz function with domain $[0,1]$ and co-domain $\mathbb R$

I would like an example of a function which is continuous with domain $[0,1]$ but is not Lipschitz continuous. Is this possible? I know a continuous function with domain $[0,1]$ is uniformly ...
1
vote
1answer
33 views

Function and Derivatives

If $$F(x)= (x-1)^{20} - (x-2)^{30} \cdot(x-3)^{40}$$ The number of real roots of $F''(x)=0$ are? $F''(x)$ - Second Derivative of $F(x)$. I have worked it out by simply differentiating it and then ...
3
votes
2answers
33 views

How to find the function for six step operation

I am trying to find a function for the following scenario: Rotating the red arrow will produce a nice sine wave as illustrated to the right of the hexagon. But I need to rotate the blue arrow, and ...
1
vote
1answer
110 views

Let $f\colon [0,1]\to\mathbb{R}$. Prove there exists $c$ such that $f(c)=c$

Let $f\colon [0,1]\to\mathbb{R}, 0<f(x)<1$ for all $x \in[0,1]$. $f(x)$ continuous. Prove there exists $c$ such that $f(c)=c$ My attempts: As $x \in[0,1]$ then $0\leq x \leq 1$ so $0 < x ...
0
votes
2answers
56 views

How to show that $f_{n+1} \leq f_n$?

Let $$f_n(t) = \frac{e^{nt}-e^{-nt}}{e^{nT}-e^{-nT}}$$ be defined on $t \in [0,T]$. Can someone help me, I need to prove that $f_{n+1}(t) \leq f_n(t)$ for every $t$. I tried taking ratios and/or ...
0
votes
1answer
29 views

If a function $f$ is invertible can I say that $f^{-1}$ is also one to one and onto?

If we have a function $f$ that is both one-to-one and onto (so it's invertible). Its inverse function $f^{-1}$ is also one-to-one and onto? If this is not true can someone please explain it to me or ...
1
vote
2answers
30 views

Question about functions and topology

This is a very general question, but one that I have been struggling with. If we say that a function from a topological space X to a topological space Y is ONTO, then does that mean that for each open ...
0
votes
0answers
18 views

How to rotate a 2nd derivative gaussian function?

I have a 2nd gaussian derivative in y and a normal gaussian in x, which results in the function: $$ f\left ( x,y \right ) =\frac{- \exp ^{-\left (\frac{x^{2} +y^{2}}{2\cdot \sigma^{2} } \right ...
0
votes
1answer
24 views

“Positively homogeneous of degree zero”

I am trying to understand a statement in an economics paper (and this paper is unfortunately quite sloppily written). Let $A$ be a finite set. Let $S$ be the set of real-valued functions on $A$, i.e. ...
0
votes
0answers
38 views

All primes in the form 4x + 1 can be written as a sum of two squares. [duplicate]

Because all primes other than 2 are odd, one of the two perfect squares must be odd, with the other being even. Is there any way to prove the statement, or is it just an observation?