Tagged Questions

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

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0
votes
0answers
6 views

A question on logic and some functional inequalities

Suppose that I have a (generic) function $g$ and arguments $a, b$ (satisfying the inequality $g(b)/b < g(a)/a$) for which I know that the following implications are true: $$g(a) < g(b) ...
0
votes
2answers
29 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
41 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
11 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
29 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
18 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
24 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
40 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
13 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
29 views

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

Are all single-valued functions bijections? If not, please explain why.
2
votes
4answers
44 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
70 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
12 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
16 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
27 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
16 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
49 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
40 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
21 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
13 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
58 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
11 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
15 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} ) ...
0
votes
2answers
37 views

Homeomorphism: Subspace to the usual topology

So, this is in reference to a question I asked earlier. "I have to show that the following function f:(0,1)→ℝ. I will use this function: $f(x)=\frac{1}{x}+\frac{1}{x−1}$." I've figured that this is ...
0
votes
1answer
17 views

Minimizing an open box (Calc I)

A rectangular container with an open top is to have a volume of $12 \;\text{m}^3$. The length of its base is twice the width. Material for the base costs (in dollars) 10/$\text{m}^2$. Material for ...
0
votes
0answers
34 views

Requesting information on constructed discontinuous functions (from any perspective)

Suppose $f:\mathbb{R}\rightarrow \mathbb{R}$ is a continuous function. Define the function $F:\mathbb{R}\rightarrow \mathbb{R}$ as $$F(x)=f(x)\prod_{n=1}^\infty\frac{x-\frac{1}{n}}{x-\frac{1}{n}}$$ I ...
0
votes
1answer
27 views

Function Relations

Suppose a function $f : A → B$ is given. Define a relation $∼$ on $A$ as follows: $a_1 ∼ a_2 ⇔ f(a_1) = f(a_2)$. a) Prove that $∼$ is an equivalence relation on $A$ b) Since $∼$ is an equivalence ...
0
votes
1answer
28 views

How do I parametrise this expression

I have no idea how to deal with the Mins when attempting to parametrise this. How do I do it?
0
votes
2answers
15 views

2 and/or 4 belongs to f(x) image?

$$f(x) = \frac{x}{x^ 2+x}+4$$ How can I now if 2 and or 4 belongs to the image of the function? (in a General Way so I can apply it to other functions)
-1
votes
0answers
23 views

$f(n - k) = \theta(f(n))$, for positive constant $ k > 1$ [on hold]

Prove/disprove: There is exist function $f(n)$, s.t. $f(n - k) = \theta(f(n))$, for positive constant $k > 1$. thanks all.
1
vote
5answers
82 views

Let A be a set of all infinite sequences consisting of 0's and 1's. Prove that A is not countable.

Sequences such as {010101010101...., 10100100100...., etc} if i am not wrong these sequences can represent all the real numbers in the binary format, so a such a set will not be countable. but i am ...
1
vote
4answers
81 views

Show bijection from (0,1) to R

I have to show that the following function $f: (0,1) \rightarrow \mathbb{R}$. I will use this function: $f(x)=\frac{1}{x}+\frac{1}{x-1}$. To show 1-1, I am using $f(x_1)=f(x_2) \Rightarrow x_1=x_2$, ...
1
vote
1answer
26 views

Identifying sets

I keep seeing written in texts the phrase 'identify sets', for example: Identify $A$ as a subset of $F(A)$ by $a\mapsto f_a$, where $f_a$ is the function which is 1 at a and 0 elsewhere. This is in ...
0
votes
1answer
17 views

Prove or disprove: functions

For a subset $X$ of $Y$, $f^{-1}(X') = (f^{-1}(X))'$ Is this true? If so, what would be a proof for it? If it isn't, what would be a counterexample? I'm completely lost The $'$ denotes "not" or the ...