Tagged Questions

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

learn more… | top users | synonyms (1)

0
votes
0answers
5 views

Extension Theorem of twice continously differentiable functions?

Is there a theorem which guarantees me that any function $f$ with bounded first and second order derivatives defined over a compact interval of $\mathbb{R}^2$ can be extended to a twice continously ...
2
votes
1answer
48 views

How can I write a function like this?

I need to write down and use a function which looks like this. It is some kind of a sinus function. I've no idea how this function looks like, and that is the reason I am looking for your help. Thanks ...
0
votes
2answers
11 views

Function bijective proving.

Let $\mathbb{C}$ be the set of all complex number. $z\in \mathbb{C}$ Given a function $$ f : \mathbb{C} \to \mathbb{C} $$ $$f(z) = (1+2i)z+5i$$ Prove that it is bijective. First, prove ...
0
votes
1answer
23 views

Closeness of set for not everywhere continuous function

I have a function $w:\mathbb{R}^2\backslash\{0\}\rightarrow\mathbb{R}$ where $w(x)\in[0,2\pi)$. I am also given for free that $w$ is continuous on $\mathbb{R}^2\backslash\{(x_1,0)\mid x_1\ge0\}$. I ...
1
vote
5answers
44 views

Linearity of a function.

I am requested to determine wether these functions are linear or not; to do that, I've to verify both the necessary conditions that are: $f(x+y) = f(x) + f(y)$ $f(\alpha x) = \alpha f(x)$ Now, my ...
6
votes
1answer
424 views

Does any one-to-one function satisfying this condition exist?

Does there exist a one-to-one function $f: \Bbb R \to \Bbb R $ such that $f(x^2) - (f(x))^2 \geq \frac 1 4 \forall x \in \Bbb R$ ? I've tested this with many one-to-one functions but the ...
3
votes
2answers
48 views

Existence of a differentiable function $f$ such that the set of points at which $|f|$ is differentiable is not dense in $\mathbb R$

Does there exist a differentiable function $f:\mathbb R \to \mathbb R$ such that the set of points at which $|f|$ is differentiable is not dense in $\mathbb R$ ?
4
votes
3answers
54 views

$f:\mathbb R \to \mathbb R$ be twice differentiable , $f(x)+f''(x)=-xg(x)f'(x) , g(x) \ge 0 , \forall x \in \mathbb R$ , then $f$ is bounded?

Let $g:\mathbb R \to [ 0,\infty)$ be a function and $f:\mathbb R \to \mathbb R$ be a twice differentiable function such that $f(x)+f''(x)=-xg(x)f'(x) , \forall x \in \mathbb R$ , then is it true ...
0
votes
1answer
16 views

Prove $|A| \leq |B|$ for $1-1$ function.

Prove $|A|\leq |B|$ if function $F:A\rightarrow B$ is a $1-1$ function. I wanted to know how to prove this out of curiosity. The help is appreciated.
0
votes
1answer
55 views

Prove that $f(x) \in A$ if and only if $x \in f ^{−1} (A)$.

Is there even a proof for this or is this just by definition : $f(x) \in A$ if and only if $x \in f^{−1}(A)$.
0
votes
1answer
20 views

Proving: If a function is bounded, then the fuction's limit is bounded.

The question I have to answer is the following: Let I be an open interval that contains the point c and suppose that f is a function that is defined on I except possibly at the point c. If $m \le ...
0
votes
0answers
14 views

A question on logic and some functional inequalities

Suppose that I have a (generic) function $g$ and arguments $a, b \in \mathbb{N}$. I know that $g$ satisfies the inequalities $$1 < \frac{g(b)}{b} < \frac{g(a)}{a} < 2.$$ I also know that ...
0
votes
2answers
31 views

Is the bijectivity of a function equivalent to monotony and continuity?

My high-school math professor told us that in order for a function $ f $ to have a reverse it must be monotonic and continuous, but I always thought that necessary and sufficient condition for a ...
0
votes
1answer
43 views

What's the name of this function?

Does the function $f(x)=\log(-\log(x))$, $x\in(0,1)$ has a name? Equivalently, the function $g(y)=f^{-1}(y)=\exp(-\exp(y))$, $y\in{\mathbb R}$. The only thing I want to know if whether this function ...
0
votes
2answers
15 views

Increasing function non-continuous on points of sequence - construction

How to construct strictly increasing function $f$, non-continuous on points of countable sequence of numbers $a_n$?
0
votes
2answers
31 views

How to find for which real numbers $a$ and $b$, the following functions are differentiable at $0$?

I need to find for which real numbers $a$ and $b$, the following functions are differentiable at $0$: $$f(x)=\begin{cases} ax+b & x < 0 \\ x−x^2 & x \geq 0 \end{cases}$$ ...
1
vote
2answers
21 views

Intersection of trig function

There are two trig function graphs on the same set of axis. $f(x)=\sin(2x)$ and $g(x)=\cos(3x)$. How do I go about finding the points of intersection of the two graphs?
0
votes
2answers
28 views

How to find if $k(x)=x^{2}\sin(\pi/2)$, $k(0) = 0$ is differentiable at 0? [on hold]

I need to find whether $$k(x)=\begin{cases} x^2 \sin \frac{\pi}{2} & x \neq 0 \\ 0 & x = 0 \end{cases}$$ is differentiable at $x=0$ or not.
0
votes
0answers
10 views

Angel function and continuity

I have the function $w:\mathbb{R}^2\backslash\{0\}\rightarrow\mathbb{R}$ given by $\cos(w)=\frac{x_1}{||x||_2}\text{ and }\sin(w)=\frac{x_2}{||x||_2}$ after some manipulation I got ...
0
votes
4answers
43 views

Number of One to One Functions [duplicate]

Suppose a set A has n number of elements and a set B has m number of elements. Then why the number of one to one functions is n!? And also, how many functions in total are possible? Are they n*m? I ...
0
votes
1answer
14 views

Finding the y-vertex of a function and X2.

I am trying to solve the following exercise: The graph of the fuction $y=-2x^2+bx+c$ passes through the point (1,0) and has as its vertex the point (3,S). What is the value of s? Options: A -5_____ ...
-2
votes
0answers
33 views

Are all single-valued functions bijections? [on hold]

Are all single-valued functions bijections? If not, please explain why.
2
votes
4answers
46 views

What function produces {0, -8, 8, -16, 16, … }?

I'm trying to figure out a function that produces the set of numbers {0, -8, 8, -16, 16, ... } when given the set of positive integers. I'm having a hard time understanding what makes some results ...
2
votes
0answers
76 views

Functional inequalities involving cubing and incrementing

Consider the set $S$ of positive increasing invertible functions $f$ satisfying: $$f((x+1)^3-1)≤(f(x)+1)^3-1$$ $$f(x^3)≥(f(x))^³$$ $$f(x)+1≤f(x+1)$$ for all positive real $x$. Clearly the identity ...
0
votes
0answers
13 views

Generic way to find codomain of a function

Is there a generic way (an algorithm maybe?) to find a codomain of a function, if the domain of all constituents is known. I.e., I have an editor where users can write simple expressions (by using ...
1
vote
2answers
39 views

How do I convert this parametric expression to an implicit one

I have: $$x=5+8 \cos \theta$$ $$y=4+8 \sin \theta$$ With $ -\frac {3\pi}4 \le \theta \le 0$ If I wanted to write that implicitly, how would I do it? I get that it's a circle, and I can easily write ...
1
vote
1answer
20 views

Define $\phi:G/H \rightarrow G/K$ by $\phi(Ha)=Ka$. Prove: $\phi$ is a well defined function.

Let $H$ and $K$ be normal subgroups of a group $G$, with $H \subseteq K$. Define $\phi:G/H \rightarrow G/K$ by $\phi(Ha)=Ka$. Prove: $\phi$ is a well defined function. [That is, if $Ha=Hb$, ...
2
votes
0answers
29 views

Function with $f(x)f(y)=f(xy)$ satisfying intermediate value property

Find all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that $f(xy)=f(x)f(y)$ for all $x,y\in\mathbb{R}$, and $f$ satisfies the intermediate value property. Taking $x=0$, we have $f(0)=f(x)f(0)$. ...
1
vote
1answer
28 views

Continuity of a function at $0$

A similar has been asked before, but it was confusing. Please help me with it. I need a general method of dealing with such problems I need to show that the following function is continuous at $0$. ...
0
votes
1answer
17 views

Find the range of a complicated function

I need to find the range of the following function : $$f(x,y) = \sqrt[4]{\frac{4x - 3y + 5}{3y-4x + 13}}$$ So my thoughts about it are first the bottom part $( 3y - 4x + 13 )$ must be greater than ...
2
votes
1answer
50 views

Weird function or not

Is $f\colon\emptyset \to\mathbb{R}$ with $f(x) = (-1)^{\frac{1}{2}}$ a function where $\emptyset$ is the empty set and $\mathbb{R}$ is the set of real numbers?
0
votes
0answers
8 views

Prove differentiability, squeeze theorem, using one-sided limits.

This is part of a larger problem, which is f(x) <= g(x) <= h(x), with f(xnull) = h(xnull), f and h are differentiable at xnull, and we eventually show that g is also differentiable at xnull. But ...
-1
votes
2answers
41 views

How to show that $f(x) = x|x|$ is differentiable at 0?

So I've gotten $$f'(x)=\dfrac{2x^2}{|x|}$$ How to show that the following function is differentiable at 0?
0
votes
2answers
14 views

What is the Term for the Center of Mass Equation Structure

What is the term for the generic structure of this form of equation: SUM(Mi * Xi) / SUM (Xi) It is the same as the center of mass calculation.
-2
votes
1answer
23 views

Determine how real parameters a,b,c are ordered [on hold]

We are told that the real-valued function $f(x) = \frac{(x-a)(x-b)}{(x-c)}$, defined except where $x=c$, will assume all real values. Can we say what is the relationship between a, b, c? E.g. is $a ...
0
votes
0answers
20 views

Determine whether a composition of functions is differentiable (prove)

Determine whether a composition of following function is differentiable $$f(x)=\sqrt{x+\sqrt{x+\sqrt{x}}}$$ Just hints, please! Thank you so much!
0
votes
1answer
17 views

An $\Bbb{R}\to\Bbb{R}$ function with two plateaus of different heights and a valley

I am looking for a $\Bbb{R}\to\Bbb{R}$ function $f$ with two plateaus of different heights and a valley. The function has a minimum for $x=a$ and $f(a)=b$. The first (the one for smaller $x$) ...
0
votes
1answer
16 views

Running time for algorithms

Suppose i have a set $\{1,2,...,n\}$ and i know that the solution to my problem is a subset $S \subseteq \{1,2,...,n\}$. Clearly trying out all subsets in an exhaustive approach is far too time ...
0
votes
2answers
22 views

Functions and its powers

Given a map $\pi: A \rightarrow B$ what is the definition of $\pi^n$ where $n$ is a positive integer? For example if $\pi(a)=b$ then is $\pi^n(a)=b^n$? Ok so if $n=3$ then ...
1
vote
2answers
59 views

for any set X, construct and injection from X to Power set of X

this is what i think, if i assume X to be {1,2,3} then P(x) will have {{1},{1,2},{1,2,3},{1,3},{3,2},{3,2,1},etc}} so will not, to say, 1 from X map to more than one element of P(x) ? so how can i ...
2
votes
3answers
26 views

addition of two differential functions is differentiable

I am stuck with proving the following statement. In fact, I am proving another assumption, and the proof of this would help me to proceed. Assume that $f_1$ and $f_2$ are differentiable on the ...
-1
votes
0answers
20 views

Absolute and relative maximum and minimum

1) $f(x)= 4x^4-17x^2+4$ The critical numbers I got are $\pm\sqrt{\frac{17}{8}}$. And 0. How do I find max and min. Rel and abs? ${}{}$
1
vote
0answers
12 views

How to change the fundamental frequency of a sample signal?

So I am dealing with a 60Hz signal that is sampled at 1kHz. This 60Hz signal has many other harmonics (eg, 120 Hz, 180Hz..... and more). For some reason, we would like it to be 50Hz. Could we ...
0
votes
0answers
15 views

Finding the value of the inverse function with inverse function theorem

I am stuck by the following problem. Let $h:\Bbb R^2\rightarrow \Bbb R^2$ and $$h(x,y)= (x^2+3xy+xy^3, x^3-5y^2)$$ Let $g=h^{-1}$ near $(0,1)$. Find $Dg(0,-5)$ I showed that the inverse function ...
0
votes
2answers
22 views

How to write $y=4x-x^{2}$ as a function with respect to $y$?

Can someone please help me write $y=4x-x^{2}$ as a function with respect to $y$? I need it to determine the volume of solid of revolution about the $y$ axis.
1
vote
1answer
17 views

Derivative relation between two equal functions

I am stuck with the following problem. Suppose $g: \Bbb R\rightarrow\Bbb R$ is $C^1$. $f(x,y)=g(x^2+y^2)$. I need to show that $xf_y=yf_x$ My attempt was: $f_x=g_x \cdot 2x$ (1) and $f_y=g_y\cdot ...
0
votes
1answer
21 views

Differentiability of two variable function with two possibilities

There is a another question which is exactly similar to my question in this website, but I think I am still confused about that too, I couldn't get it. I would be very very very thankful if someone ...
1
vote
1answer
10 views

How can I show this equality between inverses of functions?

Let $f:X\to Y$ be a function between metric spaces $X$ and $Y$. Show that for any $B\subset Y$, $f^{-1}(B^\complement)=(f^{-1}(B))^\complement$. I was able to show that they both map to ...
0
votes
0answers
6 views

Measure the affinity between two functions

I just want to know, what is the process of measure the affinity of two functions. I have no idea about if it has a proper name at all, so bear with me. For example, given two functions f(x) and ...
0
votes
1answer
21 views

easy calculus result about images of set under a function

PROBLEM: Let $f: X \to Y $ be a function and $\{ A_{\alpha} \}_{\alpha \in \Gamma}$ be a collection of subsets of $X$, then it occurs that $$ f( \bigcap_{\alpha \in \Gamma} A_{\alpha} ) ...