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

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0
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1answer
47 views

$ f(x) = -\ln\left(\tanh\frac{x}{2}\right) = \ln \frac{e^{x}+1}{e^{x}-1} $ prove $f(x) = f^{-1}(x)$

$$ f(x) = -\ln\left(\tanh\frac{x}{2}\right) = \ln \frac{e^{x}+1}{e^{x}-1} $$ Prove $f(x) = f^{-1}(x)$ when $x\gt 0$. I tried to do $f^{-1}(x) = \dfrac 1 {f(x)}$. Can you help me ?
1
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1answer
37 views

Linear algebra: function and vector space

I'm having problems with these two exercises: 1 - Functions: f(t) = t³ - 1, g(t) = t² + t - 1 and h(t) = t + 2. Is there any K in the real numbers that satisfy this condition: f(t) + k*g(t) = h(t)? ...
4
votes
1answer
75 views

How to name $x$ when $g(f(x)) = x$?

Consider two sets $A$ and $B$, and two functions $f: A \rightarrow B$ and $g: B \rightarrow A$. Assuming that $g$ is not the inverse of $f$, how should I call elements $a$ of $A$ such that $g(f(a)) = ...
-1
votes
1answer
36 views

Find the image of a homomorphism

I have the function $$\phi:\mathbb{C} \to\mathbb{C}$$ $$ z \mapsto z+3iz$$ where $C$ is the group of all complex numbers under addition, and I have to find the image and kernel. I know it's a ...
0
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1answer
76 views

What do the dots and arrows on this article mean?

I'm reading an article on how I can implement collision detection in my games. It's a really good article. However, some of the maths is confusing me: What do the dots after N mean? I've seen ...
1
vote
1answer
38 views

Has a $L^1$ bounded sequence a weak converging subsequence in $L^2$?

Let $f_n \in L^2(0,1)$ with the property that $\sup_n || f_n ||_{L^1}<A< \infty$, i.e. $f_n$ is a sequence in $L^2$ that is uniformly bounded in the $L^1$-Norm. Does $f_n$ then have a weak ...
0
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1answer
54 views

Prove or disprove: for every $f,g : \mathbb R\to\mathbb R$ even, the composition $h= f\circ g$ is even.

Proof: Given $f,g:\mathbb R\to\mathbb R$ even then $f(-x)=f(x)$ and $g(-x)=g(x)$ then $h=f \circ g$ then $h=f(g(x))=f(g(-x))$ then $h=g(-x)=g(x)$ since $x \neq -x$ the composition of two even ...
2
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2answers
29 views

Evaluating Logarithmic Expressions

Evaluate: $$\log_4 \left(\dfrac{1}{256}\right)$$ I am not sure how to approach this since there is nothing set equal to it.
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2answers
110 views

Intuitive interpretation of the alternative definition of sub modular functions

I know that the the following definition is the "usual" definition of sub modularity: $$ \forall A \subseteq B , s \notin B, F(A \cup \{s\}) - F(A) \geq F(B \cup \{ s \}) -F(B)$$ Which for me has a ...
0
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1answer
44 views

Proving equivalence of different definitions of sub modular functions

I was studying sub modular functions and was trying to prove the equivalence of different definitions of sub modular functions. Consider this first definition: $$ \forall A \subseteq B , s \notin B, ...
3
votes
1answer
114 views

Are $L^\infty$ bounded functions closed in $L^2$?

Is the set $\{ m \in L^2(0,1) : |m|_{L^\infty}\leq A \}$, (i.e. the set of $L^2$ functions with bounded $L^\infty$ norm) a closed subset of $L^2$? (Closed in the topology induced by the $L^2$-norm)
3
votes
1answer
51 views

Numbers interpreted as sets and functions

In set theory numbers are defined as sets $$\emptyset,\{\emptyset\},\{\emptyset,\{\emptyset\}\},\{\emptyset,\{\emptyset\},\{\emptyset,\{\emptyset\}\}\},\dots$$ where $n+1=n\cup\{n\}$ and ...
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2answers
39 views

Given $f(x)=x+2$ and $g(f(x))=3x^2+12x+5$. find g(x) [closed]

Please help, its for a maths investigation so my teacher cant help me. Thanks!!
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2answers
43 views

Proving noncompactnes by showing open cover with no finite subcover

Define: $$S = \{f \in C([0, 1],\Bbb R) : |f(x)| \le 1 \; \forall x \in [0, 1]\}$$ I have an open cover for the set $S$: $$U_{n} := \{f \in C([0, 1],\Bbb R): |f(0) − f(1/n)| < 1\}$$ for each $n ...
0
votes
1answer
53 views

Injection from a Well-Ordered Set to $\mathbb{Q}$

Call a set $W \subset \mathbb{R}$ well-ordered if every non-empty $A \subset W$ has a minimum element. Prove that if $W \subset \mathbb{R}$ is well-ordered, then there is an injection from $W$ to ...
1
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1answer
25 views

How to figure out if onto and $1-1$ for two cases.

I have to figure out if the function is $1-1$ and onto but I'm not sure how to do it if I have $2$ cases. $\mathbb{Z}^+ \to \mathbb{Z}^+$, where $\mathbb{Z}^+$ is the set of all positive integers. ...
1
vote
4answers
70 views

construction of a curve connecting two points

Let $a,b,c$ be positive reals numbers. Assume $a<b$. I'm trying to construct a $C^1$ function (meaning a function with continuous derivative) $f$ with the following properties: $f$ is increasing ...
0
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3answers
89 views

How would I find $f(x+h)$ for $f(x)=4-x^2$?

Can I see your work in full, too? I wanna see how you did it. I got -2x-h, but was told it was incorrect. Here's my work: $\begin{align} f(x) &= -1x^2+4x \\ f(x+h) &=-1(x+h)^2+4-(-1x^2+4)/h ...
0
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1answer
27 views

$f$ is defined and positive in $[0,1]$ for $1>a>0 \implies \inf f((a,1])>0$ if $f$ continuous on the right at $x=0$ proof that $\inf f([0,1])>0$

I have this problem : $f$ is defined and positive in $[0,1]$ for $1>a>0 \implies \inf f((a,1])>0$. if $f$ continuous on the right at $x=0$ proof that $\inf f([0,1])>0$ My proof Since ...
6
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2answers
100 views

Prove function is Fibonacci sequence

Stuck on how to finish a question I'm working on. Have to find the number of bitstrings of length n with no odd length maximal runs of ones. For example, when n=3 there are three such bitsrings: 011, ...
2
votes
1answer
80 views

The preimage of a subset

If $A\subseteq B$ under what conditions is $f^{-1}(A) \subseteq f^{-1}(B)$, where $f^{-1}$ is the preimage, not the inverse.
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1answer
52 views

Probability on $C(\mathbb{R})$

Let $C(\mathbb{R})$ be the set of continuous and bounded functions $\mathbb{R}\to\mathbb{R}$. Is there a probability measure $p$ on $C(\mathbb{R})$ such $\forall g\in C(\mathbb{R}),\ \forall ...
0
votes
1answer
20 views

How to define formula for decimal places

I need to define a formula for a half unit of the smallest decimal place in unit price ($UP$), or understand if this can be defined with a formula? What I have is this $$ T_{min,max} = Amt +/- (Qty ...
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3answers
111 views

Tough looking existence

If $a,b>0,a<b$ show that there exist $c,d\in (a,b)$ such that: $$ \left(\frac{a+b}{2}\right)^{c+d} = a^{c} \cdot b^{d} $$
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1answer
121 views

Water leaking from box and the relationship of volume and height.

Suppose we have a container that has a base of area $b$ and we fill it up with water. Volume of water = $b \cdot h$, where $h$ is height. Hence, $\mathrm{d}v/\mathrm{d}t = b \cdot ...
0
votes
2answers
26 views

Can't figure out why the domain of this function is like this?

Now, this is really solved, but i can't exactly understand why x must be ≠-1. I know this has something to do with the denominator being 0 for that value of x; but if we put "-1" in the ...
0
votes
0answers
24 views

if the function series is uniformly convergent is it also almost everywhere convergent?

Well,I know that if the series is normally or almost everywhere convergent,it's also convergent in measure.I wonder if this kind of connection is in other convergence types.For this particular example ...
2
votes
0answers
134 views

the continuity of argmin on convex funtion

Define $$x'=\text{argmin}_{x_1}f(x_1,\lambda),$$ where $f$ is a strictly convex function on $x_1$ and $\lambda$. I would like to ask if there is any theorem about the continuity of $x'$ w.r.t ...
1
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1answer
49 views

What does $C_\infty(\mathbf R)$ stand for?

The text says Cb(R) stands for the space of bounded continous functions on R. Then what does C∞(R) stand for?
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2answers
47 views

Find $F(x)=\int_{x^2}^{e^x} \cos (t^3) \ dt $

If $$F(x)=\int_{x^2}^{e^x} \cos (t^3) \ dt $$ Find $$\int F(x) dx$$ I tried to find $$\int \cos (t^3) \ dt $$ but it is not success !
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2answers
111 views

How do I solve an inequality with 2 inverse trigonometrical functions involved?

I haven't worked with this in a long time! All I remember is that increasing vs decreasing functions have the power of modifying the symbol. $$\arcsin\left(\dfrac{2}{x}\right) > ...
1
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2answers
100 views

Derivative of the following function (similar to Softmax)

I am having a hell of time trying to differentiate the following function with respect to x. Do you have any suggestions $f(x) = \frac{ w(i)^x}{ \sum\limits_{j} w(j)^x }$ where $w$ is a vector ...
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2answers
64 views

Are sine and cosine the only sinusoidal functions?

I came across sinusoidal functions while studying physics (waves and oscillations)
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2answers
56 views

Is $f(n,m) = 2n-m$ onto?

I'm having trouble determining if a function is onto. The function is $$\begin{align}\mathbb{Z}\times\mathbb{Z} &\to \mathbb{Z}\\ (m,n) &\mapsto 2n - m\end{align}$$ I know it's not one-one ...
1
vote
1answer
50 views

Determine a parameterization for the line which is tangent to the curve at t=2

(1) A curve is given by the function $$r(t)=(t^3 -3t^2 +2t +4)i + (13-5t)j +(t^2 -t-3)k$$ Determine a parameterization for the line which is tangent to the curve at $t=2$ I started by solving for ...
1
vote
1answer
37 views

Show that $g$ is not topologically conjugate to the tent function

Question: Define the function $g : [0, 1]\mapsto [0, 1]$ by $g(x)=\begin{cases}3x&\text{if}\;\;0\le x\le \frac{1}{3}\;\;\\{}\\ 2-3x&\text{if}\;\;\frac{1}{3}\le x\le \frac{2}{3}\;\;\\{}\\ ...
3
votes
3answers
49 views

Prove that there is no $(x,n,m)$ such that $f^n(x)=f^{n+km}(x)$ if $f$ is injective

Assume that $f : X\mapsto X$ is injective. And, we write $f\circ f\circ\dots\circ f(x)=f^n(x)$ for $n$ times composition. Prove that there is no $x\in X$ and no $n,m\in \mathbb{N}$ such that the ...
0
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2answers
42 views

Can the product of a monotone and a non-monotone function be monotone?

Let $f$, $g$ and $h$ be real functions of x, $x \geq 0$. Moreover, let $f(x) = g(x)h(x)$ Is it enough to know that both $f$ and $h$ are non decreasing in $x$, to conclude that $g$ must be monotone? ...
2
votes
4answers
81 views

Proving $\arctan x > \frac x{1+x^2}, \forall x >0$ with a helper function

Prove $\arctan x > \frac x{1+x^2}, \forall x >0$ There's the approach using Lagrange's, but is it also possible to define a function like so? $f(x)=\arctan x - \frac x{1+x^2}$, take the ...
0
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1answer
65 views

Creating a smooth function which is positive on some arbitrary open set $U \subset \mathbb{R}^n$.

I am looking for a $C^\infty$ function which is positive on an arbitrary open $U\subset \mathbb{R}^n$ and is zero on the boundary of $U$. Furthermore, the differential of the function on the boundary ...
0
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2answers
82 views

Prove the inverse of this differentiable function is differentiable? [duplicate]

Suppose we have a differentiable function $ g $ that maps from a real interval $ I $ to the real numbers and suppose $ g'(r)>0$ for all $ r$ in $ I $. Then I want to show that $ g^{-1}$ is ...
0
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1answer
34 views

Given $g(x)$, find $f(x)$, knowing $f(x) = \sum_{a=1}^x g(a)$

Given $g(x)$, find $f(x)$, knowing $f(x) = \sum\limits_{a=1}^x g(a)$ Is there a universal approach of finding $f(x)$, regardless of $g(x)$? For simplicity sake, assuming that $g(x)$ is a polynomial ...
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2answers
60 views

Antiderivative of Antiderivative

Probably easy but I'm not very sure. If f(x) has an antiderivative F(x) then F(x) has also an antiderivative. True or False?
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1answer
40 views

Injectivity of functions

I have these two exercises for my math-study, and I don't really know how to prove them. Can you help me out? A) Let f: X $\to$ Y and g: Y $\to$ Z be functions. Show that if g $\circ$ f is injective, ...
0
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1answer
58 views

$f$ is continuous on $[a,b]$, differentiable on $(a,b)$ , why does that imply that $g(x)=\frac {f(x)} x$?

Let $f$ be continuous on $[a,b]$, differentiable on $(a,b)$, $0<a<b$ and $\frac {f(a)} a= \frac {f(b)}b$. Why does that imply we can define a function $g(x)=\frac {f(x)} x$ and what are the ...
1
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1answer
55 views

Does $f(x,x)=0 \: \forall\; x\in \mathbb{R} \: \implies f(x,y)= g(x,y)(x-y) $?

I feel like the following results are obvious but I'm at a loss on how to prove them: $f(x,x)=0 \: \forall\; x\in \mathbb{R} \: \implies f(x,y)= g(x,y)(x-y) $ for some g $f(x,x)=1 \: ...
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vote
2answers
83 views

Proving isometries of the plane are bijective [duplicate]

So I'm trying to prove that every isometry $I:\mathbb{R}^2 \to \mathbb{R}^2$ is bijective. I have already proved that I is injective (which is almost immediate) and I also proved $I$ is continuous ...
1
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2answers
95 views

Identifying a function from its power series representation

What functions are represented by the following power series? $$\sum\limits_{k=1}^{\infty}kz^k \quad \quad \quad \sum\limits_{k=1}^{\infty}k^2z^k$$ Would this involve using a Taylor expansion? I ...
0
votes
1answer
114 views

How can one do mapping from the Cartesian product S x S to S?

If there is a finite set S, can one map S x S to S? My guess is that each element of S has an image in S x S. Am I correct? or is there a better explanation?
1
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1answer
90 views

How to find the limit of $f(x+5) - f(x-8)$, given that $\lim _{x \to \infty} f'(x)= 0$?

let $f,g$ be two differentiable functions that are defined on $\mathbb R$. its given that $g'$ is a bounded function and also given that $\lim \limits_{x \to \infty} f'(x)= 0$: show that $\lim ...