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

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3
votes
1answer
200 views

Sum of resulting values of dice

We have thrown with $n$ dice. The sum of resulting values is $k$. We are looking for a function $f$ which gives the number of throws, with we can construct $k$ with $n$ dice. Some example for ...
0
votes
2answers
44 views

How to approach questions that ask to prove a function exists?

Consider the functions $r:S\rightarrow Q$ and $h:S\rightarrow T$ for arbitrary sets $S,T$ and $Q$. Prove that: if $$r(y)=r(x)\Rightarrow h(y)=h(x) $$ then we can find a function $g:Q\rightarrow T$ ...
1
vote
1answer
79 views

Mean Value Theorem problem

Given: $f:[0, 27] \to \mathbb R$ such that, $f(0)=0$ , $f(10)=1$ , $f(27)=1$ , where $f(x)$ is differentiable. Prove that , for some $\alpha$, $\beta$ $\in(0,3)$ , the relation $$2\int_0^{27} ...
0
votes
1answer
30 views

Is there a function $f: \mathbb{R} \to \mathbb{R}$ such that $f''$ is continuous and these properties $P(f)$ hold?

By $]a, b[$ we mean an open interval. Is there a function $f: \mathbb{R} \to \mathbb{R}$ such that 1) $f''$ is continuous; 2) $f'' > 0$ on $\mathbb{R}$; 3) $f'(0) = 1;$ 4) $f \leq 100$ on $]0, ...
6
votes
3answers
444 views

How to tell where parentheses go in functional notation?

The professor gave us a function $f(z) = \ln r + i \theta$ (this is for a complex analysis class). He doesn't like answering students' questions and there's no assigned textbook so I don't know where ...
0
votes
2answers
46 views

Is $g$ equal to $g'$: injective and surjective?

So the problem says that $f: X \to Y, g: Y \to Z$, and $g': Y \to Z$ are functions. Prove that $g\circ f = g'\circ f$ being that $f$ is surjective, then $g= g'$. So I understand that $f(x) = y,\ ...
1
vote
1answer
49 views

Finding an $f(x,y,n)$ such that $round[f(x,y,n)] = \lfloor\frac xn \rfloor + \lfloor\frac yn \rfloor$

Problem: I have an equation: $$\left\lfloor\frac xn\right\rfloor + \left\lfloor\frac yn\right\rfloor$$ I need to find an equation that does NOT use the floor function, but will take those same two ...
0
votes
1answer
23 views

Help: Question About Functions

Say we want to solve an equation like $2e^{f(0)}-(f(0))^2=2$ I would like someone to explain why the following procedure is wrong. I observe that $f(0)=0$ is a solution. If $f(0)=a$ is another ...
0
votes
3answers
63 views

Suppose that $f:X \rightarrow Y$ is surjective and $A \subseteq X$ then $f(X-A) \subseteq Y-f(A)$. True or False?

Suppose that $f:X \rightarrow Y$ is surjective and $A \subseteq X$ then $f(X-A) \subseteq Y-f(A)$. I am supposed to determine whether this statement is true or false. If true I am to prove it. If ...
0
votes
0answers
29 views

Find boolean function

Given $\mathbb{B} = \{true, false\}$, and function $f: \mathbb{B} \times \mathbb{B} \times \mathbb{B} \to \mathbb{B}, f(a,b,c) = a \land b \lor c,~ \forall a,b,c \in \mathbb{B}$. I want to find a ...
1
vote
1answer
49 views

Probability density function from the inverse of another function

Given the function: $$f(x) = 1/sin(x)$$ where x is the angular interval 0 ≤ x < 1.5708 (in radians). I want to obtain a probability density function which represents the inverse case of f(x). ...
0
votes
3answers
58 views

How do I come up with a continuous function between two functions?

Say $y = 0$ when $x \leq 0$, and $y = 1$ when $x \geq 1$. I want to create a function between these two that still makes everything continuous (continuous at $x = 0$ and $x = 1$) and is monotonically ...
2
votes
0answers
54 views

Find an equation in $x$ and $k$

Find an equation in $x$ and $k$ if, $$6u-8v+2=k^2$$ $$u^{2}=1+2v^{2}$$ $$v=2xy$$ $$u=x^2+2xy-y^2$$ Since we have 4 equations, we can eliminate 3 variables. But somehow, I'm not able to find an ...
1
vote
2answers
31 views

Find the Tangent Plane (Undefined?)

I've been asked to solve for a tangent plane at a point, but the method I'm using seems to lead to an answer that is undefined. Can anyone point me in the right direction with this? Write the ...
2
votes
1answer
56 views

Let $(X,d)$ be a metric space and $f:X\to X$ a function, is $d(x,f(x))$ a lower semicontinous function?

So I was trying to prove that if $f$ satisfies a special property the the function $d(x,f(x))$ is lower semicontinous but then I couldnt come up with a counter example of the following statement: Let ...
2
votes
3answers
38 views

what function fulfills these conditions? [duplicate]

So I know that if $f(x) = x^{-1}$, than $f(f(x)) = x$ but $f(x)$ is not necessarily $x$. So now, is there $g(x)$ such that $g(g(x)) \neq g(x) \neq x$ but $g(g(g(x))) = x$? If so what is it, else why ...
1
vote
2answers
64 views

Study of the first and second derivative of $\sqrt{|x^2+x|}-x$

I am not able to study the positivity of the first and second derivatives of $\sqrt{|x^2+x|}-x$ (that is, the values of $x$ for which the derivatives are positive, negative, or zero), because the ...
0
votes
3answers
223 views

Determine conditions for constants a,b,c,d so that $f \circ g$ = $g \circ f$

I have a homework problem that I don't know how to get started in: Let $f(x) = ax+b$ and $g(x) = cx+d$ Determine necessary and sufficient conditions on the constants a, b, c, and d so that $f \circ ...
3
votes
2answers
55 views

Homeomorphism from $(-1,1)$ to $\mathbb R$

I know that $f: (-1,1) \to \mathbb R$ defined by $f(x)=\tan \Big(\dfrac{\pi}2x \Big)$ is a homeomorphism . I am looking for some other homeomorphism between $(-1,1)$ and $\mathbb R$ which is not in ...
1
vote
4answers
89 views

Is there a function $f$ such that $f'(x)=2x+f(x)$?

Is there a function $f:\Bbb R \to \Bbb R$ such that $f'(x)=2x+f(x)$? I've been trying to find it by inspection but I haven't found it, so now I'm wondering if it actually exists.
1
vote
3answers
62 views

For what values of $k$ is $g(x)=x^3+kx^2+x$ one-to-one?

I need to find for what values of $k$ $g(x)=x^3+kx^2+x$ is one-to-one. I tried finding for what values it is strictly increasing and got the derivative to be $3x^2+2kx+1>0$, but I'm not really sure ...
0
votes
1answer
64 views

Let $f(x)$ be a function defined for all positive real numbers satisfying the conditions

Let $f(x)$ be a function defined for all positive real numbers satisfying the conditions $f(x) > 0$ for all $x > 0$ and $f(x - y) = \sqrt{f(xy) + 1}$ for all $x > y > 0$. Determine ...
0
votes
1answer
69 views

Proving a function by induction [duplicate]

Let $f(n)$ be the function defined by $$ f(n) = \frac 1{\sqrt{5}} \left[ \left(1+\sqrt{5}\over2\right)^n- \left(1-\sqrt{5}\over2\right)^n \right] $$ How do you prove that $f(n) = f(n+2) - f(n+1)$ ...
0
votes
1answer
129 views

(product of) uniformly convergent functions and pointwise convergence

Consider 2 sequences of real functions on $I \subset \Bbb R$: $f_n \to f$ and $g_n \to g$ uniformly. Need to prove that $f_ng_n \to fg$ pointwise on $I$ From definition I know that $\forall ...
0
votes
1answer
30 views

Proving that a transformation of a function gives a positive result

If $x$ is real and: $$p = \frac{3(x^2+1)}{2x-1}$$ Prove that: $$ p^2-3(p+3)\geq 0$$ I think this has something to do with equating the discriminant to $0$, but I'm not entirely sure I'd really ...
0
votes
0answers
40 views

Properties of functional calculus

Suppose we have a self-adjoint bounded operator $S$ on a Hilbert space $\mathscr{H}$ with the property that $||Sx||<||x||$ for each $x\in\mathscr{H}\setminus\{0\}$. Now assume that ...
0
votes
2answers
47 views

Modify tan(x) function to be sharper

On the right is -tan(x) + Pi/2 function. On the left is a function i am trying to create which is a "sharper" version of the function on the right. Any idea how ...
1
vote
0answers
70 views

A function with a bijection

Let $r:\Bbb N^*\to\Bbb Q$ a bijection and $r_n=r(n),\forall n\in\Bbb N^*$. $f_r$ is the function such that: $$\forall x \in \Bbb R, f_r(x)=\sum_{n\in I_x} \frac{1}{n(n+1)}$$ where $$I_x=\{n \in \Bbb ...
1
vote
1answer
63 views

Finding range of $f(x) = \sin^4 x\tan x + \cos^4 x\cot x$

I got to a certain step and couldn't continue. I can't fully understand the provided solution... $$ f(x) = {\sin^6x+\cos^6x \over \sin x \cos x} = {2-1.5\sin^2 2x \over \sin 2x}$$ Let$$ t=\sin2x, t ...
1
vote
1answer
39 views

Find for which value of the parameter $k$ a function is bijective

I have to draw (by hand obviously) the plot of the following function: $$f(x)= 13\ln(\frac{x}{|x+1|})-12\ln (x+x^2) +kx,$$ for $k \in \mathbb{R}$. To do so, I have to study the first and second ...
0
votes
1answer
54 views

Find for which value of a parameter $k$ a function is injective, surjective, or bijective

Let $$f(x)=\ln(x^4-x^2+1)-2\ln(x^2+1)+2\sqrt3\arctan(\frac{2x^2-1}{\sqrt3}) + kx -k\ln x;$$ $$f_{*}(x)=\ln(x^4-x^2+1)-2\ln(x^2+1)+2\sqrt3\arctan(\frac{2x^2-1}{\sqrt3}) + kx;$$ ...
-1
votes
1answer
57 views

I don't understand this proof about graphs of cubic functions

This proof on proofs.wiki shows that: All graphs of cubic functions are transformations of an odd cubic function. I have no problems with the steps. I just do not understand what is the idea behing ...
0
votes
2answers
126 views

Suppose $f(x)$ is a rational function such that $3 f \left( \frac{1}{x} \right) + \frac{2f(x)}{x} = x^2$ for all $x \neq 0$. Find $f(-2)$. [closed]

Suppose $f(x)$ is a rational function such that $$3 f \left( \frac{1}{x} \right) + \frac{2f(x)}{x} = x^2$$ for all $x \neq 0$. Find $f(-2)$.
2
votes
1answer
113 views

Polynomial maximization

If $x^4+ax^3+3x^2+bx+1 \ge 0$ for all real $x$ where $a,b \in R$. Find the maximum value of $(a^2+b^2)$. I tried setting up ...
0
votes
0answers
57 views

Finding inflection points and concavity for given function

I have to explore function $\frac{2x-4}{1+x^2}$ It contains finding definition area of function, finding roots etc. One of the points is to find inflection points and concavity of function. I cannot ...
1
vote
2answers
31 views

What is R in the following question

I'm stuck on the following question: Show that the function $f(x) = x − \ln(x^2 + 1)$ is increasing in $\mathbf{R}$ Can anyone explain what is the meaning of $\mathbf{R}$ in this context? ...
0
votes
1answer
36 views

How to apply coordinate transformations

Lets say I want to rotate a parabola by $\pi/4$ degrees counterclockwise. Wikipedia tells me a counterclockwise transformation would mean: $$ x'=x\cos t-y\sin t \\ y'=x\sin t+y\cos t $$ however ...
0
votes
2answers
40 views

Inequality with functions [closed]

Let $ f,g : [-1, 1] \rightarrow \mathbb{R} $ which fulfil concomitant the following two conditions: $$ f(x) \leq 4^{x} - \frac{3}{2} \leq g(x) ,\; \forall \; x \in [-1, 1] $$ $$ f(x) \leq ...
9
votes
2answers
432 views

if $|f(n+1)-f(n)|\leq 2001$, $|g(n+1)-g(n)|\leq 2001$, $|(fg)(n+1)-(fg)(n)|\leq 2001$ then $\min\{f(n),g(n)\}$ is bounded

The following question was proposed at MOP 2001 A function $f:\mathbb{N}\to\mathbb{N}$ is called cautious if $|f(n+1)-f(n)|\leq 2001$ for all $ n\in\mathbb{N} $. Suppose that $f,g,h$ are ...
0
votes
2answers
96 views

One to One vs. Onto

Is there a one to one function mapping the positive integers to the open interval (0,1)? Is there an onto function mapping the positive integers onto the open interval (0,1)? I'm having trouble ...
1
vote
2answers
102 views

How to plot $\frac{2x-4}{1+x^2}$

Can someone suggest me how to plot this function (without any software)? $$\frac{2x-4}{1+x^2}$$ I have found out such points as: $(0,-4)$ and $(2,0)$ I know that $y = 0$ is horizontal asymptote ...
0
votes
2answers
36 views

Replacement Set and Functions - Repeating Number Allowed? (Algebra I Simplistic Q)

Quick, simplistic question - I apoligize in advance if I am not using correct terminology. One has a replacement set for x and y of something along the lines of $\{-2,-1,0,1,2\}$. They must plug each ...
0
votes
1answer
47 views

Show that f is one-one

I'm given this problem of: Show that $f$ is one-one $$f(x) = (\ln x - 1)(\ln x - 2), \quad 0 < x < 4$$ Now I know that to show it is one-one, I just have to differentiate the $f(x)$. But why ...
0
votes
1answer
32 views

Continuity of two multiplied functions

I am doing trying to prove that $h(x)=(fg)(x)$ is continuous and just wanted to know if $$(f(X_n))(g(Y_n))=h(Z_n)$$ Where $X_n$ , $Y_n$ and $Z_n$ are sequences, and $Z_n=(X_n)(Y_n)$
0
votes
0answers
38 views

Is there a standard name for the functions $f(x) = |x|^q$?

Do the following sets of functions have common/standard names? $f(x) = |x|^n$ for $n \in \mathbb{N}$ (or $\mathbb{N}\cup \{0\}$) $f(x) = |x|^n$ for $n \in \mathbb{Z}$ $f(x) = |x|^r$ for $r \in ...
4
votes
1answer
273 views

When does $f:X \to X$ give $L_f(A) \subseteq R_f(A)$ for all $A \subseteq X$?

Let $f:X \rightarrow X$ be a continuous function on a topological space $X$. Under what conditions is it the case for every subset $A \subseteq X$ that $$L_f(A)=A \cap \bigcap_{i=1}^{\infty} ...
1
vote
2answers
31 views

Exponential decay function multiplied by a lineair function

I have the following two function, one is exponential and the other linear. When i multiply them shouldn't there be an optimum? (So basically a minimum)
1
vote
1answer
73 views

Permutation(?) mapping [duplicate]

Problem Statement: Let $G$ be a finite group, say a group with $n$ elements, and let $S$ be a nonempty subset of $G$. Suppose $e \in S$, and that $S$ is closed with respect to multiplication. ...
0
votes
0answers
28 views

Necessary conditions on $x$ for $e^{ax} \geq b - cx$.

Let $a,c>0$, $b>1$ be constants. I am wondering what kind of necessary conditions we can find in order that $$ e^{ax} \geq b - cx$$ holds. So I would like something like $$ e^{ax} \geq b ...
2
votes
3answers
589 views

What does $R \rightarrow R$ means in functions?

I have a function. The function is: $$ f:R \rightarrow R $$ $$ f(x) = x^3$$ What does $R \rightarrow R$ means? I don't know what types of questions should I ask here. If it is not ok, the I will ...