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

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2
votes
2answers
29 views

Opposite of a function being bijective?

A function is bijective if it is both surjective and injective. Is there a term for when a function is both not surjective and not injective?
0
votes
2answers
21 views

Can a surjective function have an element in the domain not mapped to the codomain?

I have seen a lot of definitions for surjectivity stating that every element in the codomain must be mapped to something in the domain. But does the opposite also have to hold true for a function to ...
1
vote
1answer
14 views

Which function will fit this curve best?

I am trying to do a test of normality on this data set here. My QQ Plot looks like this . It looked like an arctan function to me. So my idea was to do a reverse "tan" function transformation on it. ...
0
votes
3answers
51 views

let $f$ and $g$ be two functions from $[0,1]$ to $[0,1]$ with $f$ strictly increasing. Which of the follwing is true?

let $f$ and $g$ be two functions from $[0,1]$ to $[0,1]$ with $f$ strictly increasing. Which of the follwing is true? (a). If $g$ is continuous, then $f\circ g$ is continuous. ...
3
votes
1answer
33 views

How I can simplify this inequality or how I can solve it?

How I can simplify this inequality or how I can solve it: $$\left\lceil\dfrac{\ln(t+2)}{\ln 2}\right\rceil-\left\lfloor\dfrac{\ln(t+1)}{\ln2}\right\rfloor>1$$ where $t$ is a positive integer. ...
0
votes
0answers
14 views

Existence and uniqueness of a maximum

Consider $\alpha \in [0,1]$, $\beta>0$, $\delta \geq 0$. Let $1\{...\}$ be the indicator function taking value 1 if the condition inside is satisfied and zero otherwise. Let $$ f(x,y;\alpha, \beta, ...
0
votes
1answer
15 views

Given $f(x)$ and $g(x)$ find the following and state if the composite function exists

This is a two part question. I want to find the following and determine whether or not the composite functions exist. I'm fairly sure of my functions, but I would like to confirm that they are correct ...
1
vote
1answer
16 views

Given the graphs of $y=f(x)$ and $y=g(x)$, sketch the graph of $y=k(x)$

I think I have solved this problem correctly, but I am a little unsure of whether or not the asymptotes I found are correct. My apologies for the picture, I realize it is a little small for clear ...
5
votes
1answer
53 views

Given $f(x)$ and $g(x)$, find $(fg)(x)$

I've attempted to solve the problem below, and here is what I got for a solution: Given $f(x)=x^2-9$ and $g(x)=x^2+3x-1$, find $(fg)(x).$ $$ \begin{align} (fg)(x)&=(x^2-9)(x^2+3x-1)\\ ...
0
votes
1answer
17 views

Derivation: How do I derivate this

How do I deveriate the following expression? The problem I have is the n in d^n. This expression is part of a bigger task of mine : Show via complete induktion that is true for all n from ...
1
vote
1answer
17 views

Given $f(x)$ and $g(x)$, find the following and state any restrictions

Given $f(x) = 3/(5-x)$ and $g(x) = 2x-1$ find the following and state any restrictions i) (f(g(x)) ii) (g(f(sqrt2)) Here what I got for part i: $g(x) = 2x-1$ ...
3
votes
2answers
91 views

Confusion about the definition of function

Yesterday I was talking to one of my friends about the definition of function. The formal definition of function is given by Cartesian Products but my friend's question was whether it is possible to ...
3
votes
0answers
39 views

Number of functions from domain to codomain

Let A and B be finite sets. Let a be the size of A. Let b be the size of B. Assume 0 < a < b. (a) How many functions are there with domain A and co-domain B? (b) How many one-to-one functions ...
20
votes
4answers
2k views

What is the “fastest” increasing function that's useful in some area of math?

Context: I just completed the first quarter of an Intro to Real Analysis class, and while I was thinking about how some functions (like $x^2$) aren't uniformly continuous because they, roughly ...
0
votes
4answers
19 views

Find the domain, co-domain and range of a function

The function is $$g:\Bbb R\setminus\{0\}\to\Bbb R\setminus\{1\}\;,$$ where $$g(x) = x-\frac1x\;.$$ Please pardon my formatting as I am new to this. I know what a function is of course and their ...
1
vote
2answers
29 views

Existence of continuous functions $f,g:(0,1) \to (0,1)$ such that $f\big((0,1)\big)=(0,1)$ ; and what if $(0,1)$ replaced by $[0,1)$ ?

Does there exist continuous functios $f,g:(0,1) \to (0,1)$ such that $f\big((0,1)\big)=(0,1)$ \ $g\big((0,1)\big)$ ? The problem I am having is that since $(0,1)$ is not compact I am not able to tell ...
0
votes
0answers
22 views

Sketch the graph of $y=(g-f)(x)$ given the graphs of $g(x)$ and $f(x)$

Sketch the graph of the combined function of $y=(g-f)(x)$ for the following functions of $f(x)$ and $g(x)$. I've attempted the problem already (see attached work), but I am unsure of whether I ...
0
votes
0answers
19 views

Does $t^2y''-ty'+y=2t$ with $y(t)=tz(t)$ only if $z'$ as $t^2u'+tu=2$?

I have this question: We have $y$ and $z$ functions with reals as $y(t) = t\times{z}(t)$ for $t \in I = ]0,+\infty[$. Then $y$ satisfies $t^2\times{y}''-t\times{y}'+y=2t$ on $I$ if and only ...
1
vote
2answers
27 views

Use the graph of f(x) and g(x) to evaluate (f+g)(1)

Use the graphs of f(x) and g(x) to evaluate the following. I've done what I think is all the work for this problem, but I'd like to make sure I'm on the right track with this. Feedback on whether ...
0
votes
3answers
47 views

How to come up with bijections?

Is there a good technique for finding bijections in general? Like between the integers and the natural numbers or between [0,1) and (0,1).
2
votes
2answers
74 views

How the cardinality of $\mathbb{R^+}$ and $\mathbb{R}$ same?

Let me first confirm you that this question is not a duplicate of either this, this or this or any other similar looking problem. Here in the current problem I'm asking to disprove me(most probably ...
0
votes
1answer
30 views

Are $\Re(z)$ and $\Im(z)$ solutions of $z' = az$?

I'm having trouble with a question (I have to answer "true" or "false" and explain it): We have $a:I \to C,$ a continuous function on $I$, interval of $R$. If the function $z:I \to C$ is a ...
1
vote
1answer
35 views

The function f is defined as follows: $f:A \to A$

The function f is defined as follows:$f:A$ to $A$ where$$ f(x)=\frac{3(x +1)}{x^2-1}$$ Along my proof in showing that show that there exists an x ∈ A with $f(x) = y$ (showing f is onto) ,I ran into ...
2
votes
2answers
36 views

Equality of functions

How a function which is not defined for some value can be equal to a function which is defined for the same value? How is $f(x) = \frac{(x-2)(x-3)}{(x-2)(x-4)}$ equal to $g(x) = \frac{(x-3)}{(x-4)}$ ...
1
vote
3answers
42 views

Is $g$ the unique function with this property?

Prove/Disprove: Let $A$ and $B$ be sets and let $f : A \to B$ be a function. If there is a function $g : B \to A$ such that $g\circ f = \operatorname{id}_A$, then $g$ is the unique function with this ...
0
votes
0answers
12 views

CONFIDENCE LEVEL for Median Interval

A firm wants to estimate the unknown median, m , of the height of their employees. Random Simple Size = 90 $X_{i}$ is the order statistics of the Sample Size X where height of each employee was ...
1
vote
3answers
35 views

Does the method for finding the intersection of 2 single variable functions work for multivariable functions?

I have $2$ multivariable functions $Q(x,y)$ and $P(x,y)$, I was wondering if finding the point of intersection between these 2 functions is as easy as making $Q(x,y) = P(x,y)$ as you would do for most ...
0
votes
1answer
18 views

Linear Algebra - Inner Products, Functions, and Closet Polynomial

This is the question: Formulate the linear algebra problem of finding the closet poly $p \in span \{1, t^2\}$ to the function $f(t)=e^tcos(t)$ with respect to the L$^2$ inner product: $\lt f,g\gt ...
0
votes
1answer
29 views

Can you raise trigonometric functions to a non-integer power?

I don't inmediately see any reason why you could not yet I have never come across it. For any answer given reasoning would also be appreciated! Thank you
3
votes
1answer
42 views

How do I prove this function doesn't exist?

Let $g: S\rightarrow S$ be a function such that $g$ has exactly two fixed points, and $g\circ g$ has exactly four fixed points. Prove that there is no function $f:S\rightarrow S$ such that $g=f\circ ...
0
votes
1answer
30 views

Solving ODE for x instead of y

Find the general solution of the ODE. Give the largest interval over which the general solution is defined. Determine any transient terms in the general solution. $y dx - 4(x+y^6)dy = 0$ This is ...
2
votes
1answer
52 views

The “sin-cos-maximum” function

Is there some specific notation for the function $f(x):=\max\{\cos(x),\sin(x)\}$, or maybe some equivalent compact expression? Improvement: Actually, maybe a compact equivalent expression for its ...
1
vote
0answers
25 views

Existence and uniqueness of a function generalizing a finite sum of powers of logarithms

I hope to find a proof of the following conjecture: $(1)$ For every $a>0$ there is a convex analytic function $f_a:\mathbb R^+\to\mathbb R$ such that: $f(1)=0$ and $\forall x>1,\ ...
2
votes
2answers
45 views

Prove that the function: f: $\mathbb{N} \mapsto \mathbb{Z}$ defined as f(n)= $\frac{(-1)^n(2n-1)+1}{4}$ is bijective.

Prove that the function: f: $\mathbb{N} \mapsto \mathbb{Z}$ defined as f(n)= $\frac{(-1)^n(2n-1)+1}{4}$ is bijective. This is rough. I've been staring at this one for a while now. I get stuck on the ...
-1
votes
0answers
41 views

List of simple, common functions with an incomplete domain or range on $\mathbb{C}$

This may seem like a strange question, but it's an interest of mine and I would appreciate the help of the community in addition to brainstorming on my own. As the question states, I'm looking for ...
-2
votes
0answers
32 views

algebra question MATH [on hold]

Find the indicated function and write its domain in interval notation. m(x) = , n(x) = x + 3, (m n)(x) = ? A) (m n)(x) = ; domain: [-5, ∞) B) (m n)(x) = (x + 3); domain: [-2, ∞) C) (m n)(x) = ...
0
votes
1answer
41 views

Find conditions on $m$ and $n$ that ensure that $f$ is a bijection.

Given: $X=\{0,1,2,...,m-1\}$, $f:X\to X$, $f(x)=nx \pmod m$. Find conditions on $m$ and $n$ that ensure that $f$ is a bijection. Progress It seems that $(m,n)=1$ but I can't prove that. I tried ...
2
votes
1answer
36 views

How many distinct values of floor(N/i) exists for i=1 to N.

Say we have a function $F(i)=\text{floor}(N/i)$. Then how many distinct values of $F(i)$ will exist for all $0 \leq i \leq N$ e.g. We have $N=25$ then. $F(1)=25$ $F(2)=12$ $F(3)=8$ $F(4)=6$ ...
2
votes
2answers
81 views

Is the function $y = a^x + b$ exponential?

What exactly is an exponential function? Some of the sources at which I looked said that it's a function where the rate of change at $x$ $(f'(x))$ is proportional to the value at that point $(f(x))$, ...
0
votes
0answers
39 views

Let $f:\Bbb{R}\to\Bbb{R}$ be a continuous function. Which of the following sets cannot be the image of $(0,1]$ under $f$ [duplicate]

Let $f:\Bbb{R}\to\Bbb{R}$ be a continuous function. Which of the following sets cannot be the image of $(0,1]$ under $f$? A. $\{0\}$ B. $(0,1)$ C. $[0,1)$ D. $[0,1]$ I think ...
0
votes
1answer
114 views

How many $f(x)$ are possible satisfying $f(x)=f'(x)$ and $f(0)=f(1)=0$.

Let $f:[0,1]\to\Bbb{R}$ be a fixed continuous function such that $f$ is differentiable on $(0,1)$ and $f(0)=f(1)=0$. Then the equation $f(x)=f'(x)$ admits how many solutions? The only solution ...
2
votes
0answers
47 views

How to show that $f$ can only have at most one root in $(a,b)$ with these conditions?

Let $f: [a,b]\rightarrow\mathbb{R}$ be a differentiable function on $(a,b)$. Suppose $f$ has the following property: If for an $x \in (a,b)$, $f(x)=0$, then $f'(x)>0$. The excercise is to show, ...
0
votes
1answer
28 views

A question about sets of limit points of continuous functions.

Let $f:\Bbb{R}\to\Bbb{R}$ be a continuous function and $A\subset\Bbb{R}$ be defined by $A=\{y\in\Bbb{R}:y=\lim\limits_{n\to\infty}f(x_n)$, for some sequence $x_n\to+\infty\}$. Then $A$ is ...
3
votes
0answers
40 views

How many such functions are possible?

Let $f$ be a function from $\{1,2,3,\dots,10\}$ to $\Bbb{R}$ such that ...
-3
votes
0answers
31 views

Let $A = \{1,2,3,4\}$, and $B=\{x,y,z\}$. [on hold]

a) Give an example of a function $f: A \to B$ that is onto b) Give a function $g: B \to A$ so that $f\circ g=iB$. c) Is your function $g$ one-to-one? Justify. Any help on this homework problem I ...
0
votes
2answers
20 views

Proving that the relation from the null set to the null set is a function

How would one prove that a relation that maps the null set to the null set is a function? I tried showing that the domain of the relation is the null set, but I'm unsure of where to proceed from ...
1
vote
2answers
36 views

Order of $f(n) = 4n + 6n^3 - 8n^5$

If a function $$f(n) = 4n + 6n^3 - 8n^5$$ then the order of $f$ is: The answer I have is $\log(n)$, but I'm not sure if it's right.
1
vote
4answers
28 views

Find the limit of function using Taylor series

Good evening, I'm somehow stuck on solving some easy exercises : $$\lim_{x\to\infty} x^{3/2}\bigl(\sqrt{x+1}+\sqrt{x-1}-2\,\sqrt{x}\bigr)$$
0
votes
1answer
34 views

$C^l$ diffeomorphism between a smooth manifold and a $C^k$ manifold

Let $M$ and $N$ be two Riemannian manifolds. $M$ is smooth while $N$ is $C^k$ manifold. Suppose there is a $C^l$ diffeomorphism between the two manifolds for $l \leq k$. Is it true that $N$ is also ...
-3
votes
0answers
52 views

Bijection from $\mathbb{Z}$ to $\mathbb{Q}$ [on hold]

Can you explicitly tell me a bijection from $\mathbb{Z}$ to $\mathbb{Q}$. I need an explicit one. Thanks in advance.