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

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1answer
82 views

Show that the identity function over any set is a bijection

Let $A$ be any set, and let $I: A\to A$ be the identity function on $A$. To show this identity function over $A$ is a bijection. We can show that it is injective and surjective. We can readily know ...
3
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3answers
140 views

Given $Re\{f'(z)\}$, to find $Im\{f(z)\}$

An analytic function $f(z)$ is such that $\Re\{f'(z)\} =2y$ and $f(1+i)=2$. Then the imaginary part of $f(z)$ is $-2xy$ $x^2-y^2$ $2xy$ $y^2-x^2$ Here, by using Milne Thomson's ...
1
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1answer
38 views

Can we find two different real functions f and g such that f is a composition of g's and g is a composition of f's?

Can we find two different real functions $f$ and $g$ such that $f$ is a composition of $g$'s and $g$ is a composition of $f$'s? ($f,g \; \mathbb{R} \rightarrow \mathbb{R})$. My ideas: Well, if f is a ...
2
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1answer
41 views

For any strictly increasing convergent sequence $x_n$, the sequence $f(x_n)$ is convergent

Let $f(x)$ be defined on R and be strictly increasing. Claim: for any strictly increasing convergent sequence $x_n$, the sequence $f(x_n)$ is convergent. I believe it's false. Think about ...
1
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1answer
55 views

What kind of function may satisfy this set of properties?

What's an explicit, expressible function that I can program on a computer using basic arithmetic that satisfies the following properties: $$ f \text{ is differentiable}\\ f(0)=0\\ f(1)=1\\ ...
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0answers
41 views

Prove that if a function f is continuous then it is monotonic

Let ($L_1 , \le_1 $) and ($L_2 , \le_2 $) be two posets. A function $f : L_1 \to L_2 $ is continuous if $ \forall E_1 \subseteq L_1, E_1 \neq \emptyset, f(LUB(E_1)) = LUB(f(E_1)) $ Prove that if $f$ ...
1
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5answers
85 views

Suppose that $A$ and $B$ are sets and that $f: A \rightarrow B$ is onto. Does being onto guarantee the sets are finite?

Suppose that $A$ and $B$ are sets and that $f: A \rightarrow B$ is onto. Determine which of the following statements are true: If $A$ is finite then $B$ is finite. If $B$ is finite, ...
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0answers
19 views

Find the interquartile range of this piecewise function

$$f(t)= \begin{cases} |2-t|, & \text{$1 \leq t \leq 3$} \\ 0, & \text{Otherwise} \end{cases}$$ I have graphed the piecewise function, but I have no idea how to find its interquartile range! ...
1
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1answer
60 views

Prove that if $X$ and $Y$ are independent discrete variables, then $f(X)$ and $f(Y)$ are independent.

Prove that if $X$ and $Y$ are independent discrete variables, for $f: \mathbb{R} \rightarrow \mathbb{R}$, then $f(X)$ and $f(Y)$ are independent. Here is the exact same question. I define ...
2
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3answers
741 views

Are the sum and/or product of two increasing functions also increasing?

Question: Let $f(x)$ and $g(x)$ be two increasing functions. a) Show that their sum is also increasing. b) Investigate the corresponding claim for the product of two increasing functions. ...
1
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0answers
78 views

Determining the Domain and Range of a multi-dimensional function

$ f(x,y,z) = \frac{3}{2-x} + \frac{1}{y-z} $ i) Write down the domain of $f$ ii) Determine the range $T$ of $f$. For each $c \in T$ find a point $(x,y,z) \in \mathbb{R}^3$ such that $f(x,y,z) = 1 $. ...
2
votes
1answer
76 views

Showing this function is continuous $ f:(x,y)\mapsto x^2+y^2$

I have the following function: $$f:\Bbb R^2 \to \Bbb R,\quad f:(x,y)\mapsto x^2+y^2$$ I want to show that this function is continuous by showing that $f^{-1}((a,b))$ is an open set. How do I ...
0
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1answer
57 views

Is the following function finite?

Let $1 < p < 2$ and $\forall \ \xi \in \mathbb{R}-\{0\}$ sucht that $$\varphi (\xi) = \frac{|1 + \xi|^p - 1 - p \xi}{|\xi|^p}.$$ We have that $$\lim_{\xi \to \pm \infty} \varphi (\xi) = 1 \;\; ; ...
2
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1answer
134 views

If the period of the function $cos(nx)sin(5x/n)$ is $3\pi$ then what should be number of integral values of n?

If the period of the function $\cos(nx)\sin(5x/n)$ is $3\pi$ then what should be number of integral values of $n$ ? My approach : I tried like period of $\cos(nx)$ is $2\pi$/n and $\sin(5x/n)$ is ...
3
votes
1answer
40 views

Homomorphisms Groups Kernel

Given G: group of units in Z mod 14 under multiplication. A function sends the integers under addition to G. $f(n)$ = $[3]^n$ I am just checking whether I am correct in stating that the kernel ...
0
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0answers
34 views

Functions understanding

So lets say im Trying to find (G o F)(7) and (F o G) I want to see if im on the right track G o F = f(g(7)) = f(7-1) = (7-1)^3 = 261 F o G = g(f(7)) ...
3
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1answer
247 views

Does there exist a continuous onto function $f:[0,1] \to (0,1)$?

If there is, what's an example. If not, how do I prove none exists?
2
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1answer
28 views

How could I show that the set of degenerate critical points of a $C^{\infty}$ function is a closed subset of $\mathbb{R^n}$?

What would be a good idea to show( or finding a counter example) the set of degenerate critical points of a $C^{\infty}$ function $f: \mathbb{R^n}\to\mathbb{R}$ is a closed subset of $\mathbb{R^n}$? ...
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3answers
84 views

Solve for $n$ in $2^n=8$

So, I was wondering if it is possible to solve for $n$ in $2^n=8$ (or any other question where $n$ is a power) using $9^{th}$ grade math. Please excuse my naïveté if this is extremely stupid/simple. ...
3
votes
1answer
92 views

Inverse of $f(x) = 3x + \cos(x)$

Was hoping someone could help me find the inverse of $f(x) = 3x + \cos(x)$ The steps I took were: $y = 3x + \cos(x)$ $x = 3y + \cos(y)$ $x - 3y = \cos(y)$ $\arccos(x-3y) = y $ But I still have a ...
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2answers
63 views

What is the relation between $\nabla f(\mathbf x), \mathbf y - \mathbf x, \text{ and } f(\mathbf y) - f(\mathbf x)$?

The Problem: Let $f(\mathbf x)$ be a convex function on $\mathbb R^n$. Given two points $\mathbf x$ and $\mathbf y$, what is the relation between $\nabla f(\mathbf x), \mathbf y - \mathbf x, \text{ ...
3
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2answers
61 views

Showing that a function is surjective (onto)?

For example : $F:\Bbb R\rightarrow\Bbb R$ defined by $F(x) = \frac{2x+1}{3}$ I let $F(x)=Y$ which gives $Y=\frac{2x+1}{3}$ then simplify and solve for $x$ , what I have at the end is ...
0
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1answer
47 views

Proving the limit of the power of two functions is the power of the limits?

I've already seen a couple of times both on questions here (like Value of $\lim_{n\to\infty}{(1+\frac{2n^2+\cos{n}}{n^3+n})^n}$ or Problem of limit of power function) and in other online resources ...
1
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1answer
41 views

What does the notation $\{ \pm 1 \}^X$ in relation to functions and hypothesis classes means in the context of PAC learning over half spaces?

I was reading the following paper (on PAC learning over half-spaces) and encountered the following notation for a hypothesis class (on page 4): $$\mathcal{H} \subset \{ \pm 1 \}^X$$ However, it was ...
7
votes
5answers
180 views

Show that if $g \circ f$ is injective, then so is $f$.

The Problem: Let $X, Y, Z$ be sets and $f: X \to Y, g:Y \to Z$ be functions. (a) Show that if $g \circ f$ is injective, then so is $f$. (b) If $g \circ f$ is surjective, must $g$ be surjective? ...
3
votes
2answers
128 views

$f\left( x-1 \right) +f\left( x+1 \right) =\sqrt { 3 } f\left( x \right)$

Let f be defined from real to real $f\left( x-1 \right) +f\left( x+1 \right) =\sqrt { 3 } f\left( x \right)$ Now how to find the period of this function f(x)? Can someone provide me a purely ...
1
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1answer
50 views

Characterization of monovalued functions

Let $f$ be a binary relation. Let $(\bigcap G)\circ f = \bigcap_{g\in G}(g\circ f)$ for every set $G$ of binary relations. Can we prove that $f$ is monovalued (a function)?
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2answers
27 views

Limit of mappings over a real function F

I have read in a book that if we let $F$ be a bounded function, and $M$ a mapping, such that $F = MF$ is satisfied. Then $$F = \lim_{N \to \infty}M^NF$$ is valid. Can this be the case in general? ...
1
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1answer
46 views

Injections from a set of functions to R

Show there is an injection from $\Bbb R^2 \to \Bbb R $ Does there exist an injection from $X \to \Bbb R$ where $X $ is the set of all functions where f(x)=x for all but finitely many x. This is a ...
5
votes
4answers
430 views

Given $f(1)=10,f(2)=20,f(3)=30$ find $f(12)+f(-8)$ for a 4-th degree monic polynomial

If $f(x)=x^4+ax^3+bx^2+cx+d$. Given $f(1)=10,f(2)=20,f(3)=30$ find $f(12)+f(-8)$. This problem has troubled me a lot.The more I try to solve it,it becomes lengthier. My problem is that there are four ...
0
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1answer
54 views

Find 2 functions

I need to find two functions or rather two sequences, both of which are borderless but actually converge if you look at $min(a_n, b_n)$. The min has a limes but neither of them does for itself and min ...
4
votes
2answers
91 views

$\lim_{n\to ∞} \left[\frac{f\left( x +\frac1n\right)}{ f(x)}\right]^n$

Could anyone solve this problem for me? Let f be a positive differentiable function on the internal $\left[\,0,\infty\right)$. $$\lim_{n\to ∞} \left[\frac{f\left( x +\frac1n\right)}{ ...
0
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0answers
31 views

Combine several different equations with restrictions, without using piece-wise functions.

Note: this is not the same as my other question, found here. How do I combine several different equations, such as: ...
0
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1answer
172 views

Promissory note example Financial Math

Brenda owes Cathy $\$8500$ and has signed a promissory note to repay the debt in 15 months from the signing date. The note was signed on December 6, 2009, and the maturity value of the note is ...
1
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1answer
170 views

Calculating limits of the integral for a probability density function

I'm revising for my probability exam, and I'm in the process of going over some past tutorials. I'm have severe issues with calculating the limits of the integral used to calculate marginal densities. ...
1
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1answer
135 views

Continuity of $f(z) = Log z$ , for $z$ complex, non-real $-\ln|z|$, for $z$ real

At what points in the complex plane is this function continuous (If there is any)? Would it be correct to conclude that $f$ is then continuous for all $z$ in the complex plane less the real ...
3
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2answers
124 views

Functional equation $f'(x)=cf(x+1)$ has a solution if and only if $c\leq 1/e$

In Contests in Higher Mathematics: Miklos Schweitzer Competitions, 1962-1991 by Gabor J Szekely, problem F.57 there is the study of $f~:~[0,\infty)\to (0,\infty)$ such that: $\exists c>0, \forall ...
2
votes
1answer
53 views

Conditions that guarantee a composite Bezier curve in the cartesian plane represents a function?

Context I am allowing users of my application to control a curve connecting $(0,0)$ and $(1,1)$. There are a finite number of knots that are evenly spaced horizontally. The user can specify the ...
0
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0answers
113 views

Function Mapping Definition in Relation

So I have two relations: i) Ordinary relation ii) Disjunctive relation. Definition of Ordinary relation: $\Sigma$ be a finite set of attribute names, where for any attribute name $A\in \Sigma $, ...
0
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2answers
42 views

Combine several functions with restrictions , keeping only one equation

How do I combine these two functions that both have restrictions into a single equation? $$f(x)=|x|\times\frac{\sqrt{x^2-1}}{\sqrt{x^2-1}}$$ and $$g(x)=-|x|\times\frac{\sqrt{1-x^2}}{\sqrt{1-x^2}}$$
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3answers
33 views

Restrict $x$ in an equation, but keeping only one equation

How do you put restrictions on the $x$ in an equation without writing more than one equation? This is a two part question: How to take out a section of the graph of an equation? How to take out ...
0
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0answers
39 views

finding the domain of a function raised to the power $1/1996$

$$f(x)=\left(\frac x{1-|x|}\right)^{1/1996}$$find the domain Answer:$(-\infty ,-1) \cup(0,1]$ condition Ist: $1-|x|\neq0 \implies x\neq\pm1$..............(1) condition IInd: $1-|x| \ge0 ...
1
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1answer
25 views

Non existence of one to one function

Show that there exists no 1-1 function $f:\mathbb{R} \rightarrow \mathbb{R}$ with the property \begin{equation*} \forall x \in \mathbb{R},~f(x^2) -(f(x))^2 \geq 1/4. \end{equation*} How do I bring ...
1
vote
1answer
65 views

Polynomial satisfying $f(x) = f'(x) \cdot f''(x)$

If a polynomial of degree $n$ satisfies $f(x) = f'(x)\cdot f''(x)$ (such that $n$ belongs to $\mathbb R$) then $f(x)$ is? A) an onto function B) an into function C) no such function possible D) ...
0
votes
2answers
62 views

Is there a sample of a $f(x)=y$ multivalued function whose inverse $f(y)=x$ is also multivalued?

Trying to learn about the properties of the multivalued functions, I found the definition at the Wikipedia as "a left-total relation (that is, every input is associated with at least one output) in ...
-1
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1answer
37 views

Is it possible to extend the set-inclusion order of a power set to a well-ordering?

The original aim is to define recursively a function on the power set of a set such that the functional value of a subset is determined by those of its proper subsets. Thank you.
9
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1answer
129 views

Find the function of separation between two functions

I seriously doubt that is what it is actually called, but I'm not very knowledgeable in this matter. Conceptually, what I am trying to do is calculate the function of a line/curve that shows the ...
0
votes
1answer
86 views

Prove that function has no finite limit using $\epsilon$ - $\delta$ defintion

I want to prove using $$(\exists \varepsilon > 0)(\forall \delta > 0)\exists x(0 < \left| {x - {x_0}} \right| < \delta \Rightarrow \left| {f(x) - L} \right| \ge \varepsilon )$$ That the ...
1
vote
1answer
64 views

Find the residue of the function $g(z)=f(z^2)$ at a given point.

Let $f(z)$ be analytic in $0<|z|<R$. Find the residue of the function $g(z)=f(z^2)$ at $z_0=0$. I am looking for a solution to this problem. My thoughts: I know in order to find the residue ...
2
votes
3answers
66 views

Find the inverse of $f(x,y) = (x+3y,3x+y)$

Given the function $f : \mathbb{R}^2 \to \mathbb{R}^2$ as $f(x,y) = (x+3y,3x+y)$. Find $f^{-1}$ .( Assume $f$ is a bijection) I know how to find $f^{-1} (x) = (3x+2)$ or anything with one ...