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

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2
votes
1answer
38 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
46 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
44 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
38 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
32 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
54 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
47 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
42 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
43 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 ...
3
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
86 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
40 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
115 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
48 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
41 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
34 views

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

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
21 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
37 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
35 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
53 views

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

Can you explicitly tell me a bijection from $\mathbb{Z}$ to $\mathbb{Q}$. I need an explicit one. Thanks in advance.
0
votes
0answers
24 views

Analyze the variation of $f(x)=(1+\frac{1}{x})^{-K}$ $_2F_1((K-1)a,K,Ka,\frac{1}{1+x})$ w.r.t. $x$

Is there any way to analyze the variation(w.r.t. $x$) of the following function: $f(x)=(1+\frac{1}{x})^{-K}$$ _2F_1((K-1)a,K,Ka,\frac{1}{1+x})$, where $ _2F_1$ is the Gauss' Hypergeometric Function, ...
0
votes
1answer
14 views

Are the roots of a smooth function, a smooth function?

Let $f(x,y)$ be a smooth function. It is given that for every $x$ there exists at least one $y$ such that $f(x,y)=0$. Is this possible to select one such $y$ for every $x$, such that the $y$'s are a ...
3
votes
1answer
61 views

If $f(x)+2f(1/x)=3x$, find all $y$ such that $f(y)=f(-y)$.

The function $f(x)$ is not defined when $x=0$. This function has the property that $f(x) + 2f\left(\frac 1x\right) = 3x$. Find all such values of $y$ such that $f(y) = f(-y)$. (This means it is an ...
0
votes
2answers
53 views

$y=e^{-x}$ and $y=x$ point of intersection

How can I find the point of intersection of $y=e^{-x}$ and $y=x$ ? Here's the graph
-2
votes
1answer
14 views

Value of $K$ so that $f(x)=3x^3+6x^2+KX-4$ has the same remainder when it is divided by $x-1$ and $x+2$ [closed]

Value of $K$ so that $f(x)=3x^3+6x^2+KX-4$ has the same remainder when it is divided by $x-1$ and $x+2$. Thanks for the help!
0
votes
1answer
20 views

How can I tell if the function $f(n)=2n$ on $\mathbb Z$ is one-to-one, onto, or both?

The domain of the function is the set of all integers. The codomain of each function is also the set of all integers. $$f(n) = 2n $$ I was thinking that the function is one-to-one but I don't know ...
1
vote
1answer
25 views

How to find $\lim\limits_{z \to z_0} \frac{{\overline z}^2-{\overline z_0}^2}{z-z_0}$

Fairly simple question, I want to find this limit $$\lim_{z \to z_0} \frac{{\overline z}^2-{\overline {z_0}}^2}{z-z_0}$$ The original question was to find the region at which the function ...
0
votes
1answer
21 views

Relaxing Monotonicity of a Function $f:\mathbb{Z}\rightarrow \mathbb{R}$

Suppose a function $f:\mathbb{Z_+}\rightarrow \mathbb{R}$ fails monotonicity, but not by much. For example $f(2)= .3$ and $f(z)=1/z$ otherwise. Here there exists a single point where the function is ...
0
votes
2answers
27 views

how to find a spline function from given control points

consider having an n-amount of control points in 2D space, what's the best way to find the function passing through the start and end points while approximating the path according to the other given ...
0
votes
1answer
10 views

how to prove that a given function is univalent

I have to prove that following function is univalent $f(z) = z^2 +3z +1, ~|z|<1$ in complex plane. What I tried is: Let $f(z_1) = f(z_2)$ $\Rightarrow$ ${z_1}^2 +3z_1 +1= {z_2}^2 +3z_2 +1$ ...
2
votes
0answers
17 views

Proving a basic result about Holder continuous functions

Let $V$ be a open convex set. We will say that a function $m$ has the order of smoothness $p$ on $V$ with $p=l+\gamma$, where $l \geq0$ is an integer and $0<\gamma\leq1$ and will write $m\in ...
1
vote
2answers
48 views

Bijection between $\mathbb R^2$ and $(0,1)$ [closed]

I want to prove the sets have same cardinality: $\mathbb R^2$ and $(0,1)$ Please Help.
4
votes
1answer
76 views

Suppose that $f: A \to B$ and $g: B \to C$ are functions.

Suppose that $f: A \to B$ and $g: B \to C$ are functions. Prove the following: (a) If $g \circ f$ is injective, then $f$ is injective. Proof. Assume that $f$ is not injective. Then ...
1
vote
1answer
54 views

Using continuity to prove f is a constant function

Recently missed this problem on an exam. Just went to office hours to clarify what the proper proof was and wanted to see if, in attempting to repeat the problem, I can figure out if there are better ...
1
vote
1answer
19 views

Getting a function that passes through 'n' points

Is there a way to get a periodic function that passes through n arbitrary points?
0
votes
1answer
55 views

Find $\int_2^{2.2}f(x)\,\mathrm dx$ given $f(x)=x^4-3x^3+9x^2+22x+6$.

$f(x)=x^4-3x^3+9x^2+22x+6$. Find $\int_2^{2.2}f(x)dx$ by finding $f(x-2)$ This is in a non-calculator paper which is why $f(x-2)$ is meant to be obtained (it's supposed to made the maths possible to ...
0
votes
1answer
26 views

XOR function over binary vectors

I didn't really know how to name this question, it has been bothering me for some time: You are given n binary vectors of dimension $d: x_1,\cdots,x_n$; $x_i = x_{i_1},\cdots,x_{i_d}$. You are also ...
3
votes
2answers
56 views

If a function is Riemann integrable, then it is Lebesgue integrable and 2 integrals are the same?

Is is true that if a function is Riemann integrable, then it is Lebesgue integrable with the same value? If it's true, how to prove it? If it's false, what is a counterexample?
0
votes
0answers
27 views

Computationally Plotting an Equation

I have the following equation: $y^2 - log(y)^2 = 4log(x) + \frac{4}{x} + c$ and I want to plot this, real terms only, in the eqi-axis interval 0 to 5, for different values of c. Is there a general ...
0
votes
0answers
8 views

Normalization for argument of maximum function

is it possible to normalize the maximum function of a certain argument ? Means: Is that $\theta_{ML} = arg \max\limits_{\theta} \{ \sum \limits_{n=1}^{N} |w_n w^*_{n+N}| - \Big( \frac{SINR + ...
1
vote
1answer
49 views

On consequences of $\int_{0}^1f(x)x^ndx=0 , \forall n \in \mathbb Z^+\cup\{0\}$

If $f : [0,1] \to \mathbb R$ is a continuous function and $\int_{0}^1f(x)x^ndx=0 , \forall n \in \mathbb Z^+\cup\{0\}$ then is it true that i) $\int_{0}^1(f(x))^2dx=0$ ? ii) ...
1
vote
1answer
19 views

Taylor expansion of Airy function

We know that Taylor expansion is : $ f(x_0 + h) = f(x_0) + h f'(x_0) + .. \ $ I wish to expand the Airy function about it's first root , i.e , $Ai (c_1 - \epsilon ) = Ai (c_1) - \epsilon A_i'(c_1) ...