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

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1answer
50 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
112 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) > ...
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2answers
101 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 ...
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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 ...
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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}\;\;\\{}\\ ...
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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 ...
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2answers
43 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? ...
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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 ...
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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 ...
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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 ...
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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, ...
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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 ...
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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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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 ...
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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 ...
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1answer
116 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?
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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 ...
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1answer
78 views

Generate easing function from hash table/value pairs

I have a hash-table/value-pair list consisting of, what I call, linear control value paired with curved/eased real values. Something like this: 0 = 0 1 = 0.0010000000000000002 2 = ...
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3answers
37 views

Consider a function $g: (-1, +1) \rightarrow \mathbb{R}$

Consider a function $g: (-1, +1) \rightarrow \mathbb{R}$ given by $g(x) = \frac{x}{1-|x|}$ Show that $g$ is 1-1 and find $g((-1,1))$ Find $g^{-1}$ Are $g$ and $g^-1$ continuous I found that $g$ ...
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0answers
33 views

Is there a syntax to refer to the domain of a partial function?

I am trying to formally represent a Map (in programming sense), with a partial function mapping. Lets say this partial function is $Q : S \mapsto T$, where $S$ and ...
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0answers
81 views

Using the Intermediate Value Theorem to prove a statement about an equation true

I want to prove this statement true by using the IVF: For any real number $b > 2$, the equation $2^x = bx$ has a solution. Here are some questions I need help with answering: Define a function ...
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3answers
52 views

Prove/disprove: $\forall f\ \in \mathbb N ^{\mathbb R}. \forall x\in \mathbb R. \exists y\in \mathbb R ((f(x)=f(y))\wedge (x\neq y))$

Prove/disprove: $\forall f\ \in \mathbb N ^{\mathbb R}. \forall x\in \mathbb R. \exists y\in \mathbb R ((f(x)=f(y))\wedge (x\neq y))$ This statement looks very similar to the definition of ...
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3answers
79 views

If $f\circ f\circ g\circ g\circ f\circ f$ is invertible, so is $g$ [closed]

Let $f\circ f\circ g\circ g\circ f\circ f$ be invertible (i.e., have left and right inverse functions). Prove $g$ is invertible as well. I would appreciate it if you helped me.
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2answers
1k views

Why are linear functions linear?

I always thought linear functions need to satisfy $$f(x+y)=f(x)+f(y).$$ I am a tad confused now, consider $f(x)=2x+3$. $f(1)=5$, $f(2)=7$, $f(1+2)=f(3)=9 \neq f(1)+f(2)$ which was what I thought ...
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2answers
38 views

Find the domain of a function

Find the domain of: $$\frac{1}{\sqrt{3-|5-\frac{2}{x}|}}$$ I really don't know how to start. Can anyone help? Thanks.
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2answers
24 views

Showing a function $H:\mathbb R^{\mathbb R}\to \mathbb R^{\mathbb R}$ is onto and finding its inverse

Let $H:\mathbb R^{\mathbb R}\to \mathbb R^{\mathbb R} \\ H(f)=\begin {cases}f^{-1} & \text {$f$ is a bijection} \\ f & else \end {cases}$ Prove that $H$ is onto (I already showed it's ...
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2answers
28 views

For a composition to be defined: $Domf\circ g\subseteq Dom f, Im f\circ g \subseteq Im g $?

For a composition to be defined, is the following two a must? $$f:A\to B, g: C\to D\\ f\circ g : C\to B \\ Domf\circ g\subseteq Dom f\\ Im f\circ g \subseteq Im g $$ Are there other conditionals for ...
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1answer
69 views

Does this function $ h $ exist?

If we have a function $ f $ from the interval [-1,1] into the real numbers and $ f(x)=0 $ when $ x $ is greater than or equal to -1 and less than or equal to 0 and $ f(x)= 1 $ for $ x $ greater than ...
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8answers
731 views

How to prove the function $\sin(x)$ is not onto

\begin{align} f:\mathbb{R}\rightarrow\mathbb{R},\:f\left(x\right)=\sin\left(x\right)\tag{1} \end{align} I know that it is not onto because for all values of $y$ past $[-1,1]$ there is no $x$. ...
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1answer
41 views

Number of jumps of right continuous maps with left limits.

i'm seeking a very short (and self contained) proof that given a map $$f : t\in [0,T] \to f(t)\in \mathbb{R}$$ which is right continuous with left limits, setting $$\Delta f (t) = f(t)- f(t_{-})$$ the ...
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1answer
64 views

Find all functions satisfying $(1+y)\,f(x) - (1+x)\,f(y) = y \, f(x/y) - x \, f(y/x)$

Find all functions which satisfy: $$(1+y)\,f(x) - (1+x)\,f(y) = y \, f(x/y) - x \, f(y/x)$$ for all real, $x,y \ne 0$ and which takes the values $f(1) = 32$ and $f(-1) = -4$ I am not sure, which ...
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1answer
289 views

Constructing a bijection between $[a,b)$ to $(0,1)$

Construct a bijection between $[a,b)$ to $(0,1)$ I'll be using a technique that builds the function in steps from one interval to the next: $[a,b)\overset{x-a+1}\to[1,b-a+1)\overset{\frac 1 ...
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1answer
13 views

Linking summations with their correct function(s)

Guys can you please guide me step by step on how to link given functions with the functions to choose from. So for example a function $g(n)\in \Theta n^2$ and if there is no match then you say there ...
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1answer
31 views

Ways to prove that $|A|\neq |\mathcal P(A)|$ with CSB

Prove that $\forall A$, $A$ is a set: $|A|\neq |\mathcal P(A)|$ with CSB (Cantor–Schroeder–Bernstein theorem). I'm reading a proof that show that there's no surjection between $A\to \mathcal ...
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0answers
23 views

Proving $\forall f\in \mathbb R ^{\mathbb R}(f\neq i_{\mathbb R})\to (\exists g\in \mathbb R ^{\mathbb R} (f\circ g \neq g\circ f))$

Prove $\forall f\in \mathbb R ^{\mathbb R}(f\neq i_{\mathbb R})\to (\exists g\in \mathbb R ^{\mathbb R} (f\circ g \neq g\circ f))$ Proof by contradiction: For all $f(x)\neq x$ and $(\forall g\in ...
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1answer
320 views

Find a bijective function between two sets [duplicate]

I want to find a bijective function from $(\frac{1}{2},1]$ into $[0,1]$. So, What is a bijective function $f:(\frac{1}{2},1]\to[0,1]$?
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1answer
50 views

$f(x)$ is onto?

A function $f:R-{a_1,a_2}$ to $R$ is defined by $$f(x)=\frac{ Ax^2+6x-8}{A+6x-8x^2}$$ How many integral values of $A$ exist for which $f(x)$ is onto. I tried finding the range of this function, but I ...
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1answer
63 views

Determine value $b$ in $f(x)=ab^x$ given the following data points

If $f(x)=ab^x$, what is the value of $b$ if $(0,35)$ and $(3,125)$ are data points? Is this the way to do it? $$35=ab^0,$$ $$a=35.$$ $$125=ab^3,$$ $$125=3\log(35)+\log(b),$$ ...
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1answer
168 views

Combination Problem: Girls picking flowers

Two girls have picked 10 roses, 15 sunflowers and 15 daffodils. What is the number of ways they can divide the flowers amongst themselves ? ...
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1answer
28 views

Problem on Function Mapping

Let $S = \{1, 2, 3,... m\}$, $m > 3$. Let $X_1......X_n$ be subsets of S each of size 3. Define a function f from S to the set of natural numbers as, f(i) is the number of sets $X_j$ that contain ...
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0answers
63 views

Application of Stone Weierstrass theorem

Let $K$ be the unit circle in the complex plane and let $A$ be the algebra of all functions of the form $f(e^{i\theta})=\Sigma_{n=0}^{N}c_{n}e^{in\theta}$ ($\theta$ real). Prove that $A$ is uniformly ...
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1answer
66 views

Comparing with a sequence function

I have a big data set of positive integers, inside it, I found a sequence of integers $k$ that looks like that $\{k_1,k_2...k_i\}$ Accordingly I have $f(n)$ $f(1)= k_1$ $f(2)= k_2$ $f(3)= k_3$ ...
1
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0answers
56 views

Can uniform continuity of a differentiable function be formulated only in terms of limits or derivatives?

Reading this, this and this Q&A's I've understood that a uniformly continuous differentiable function on $\mathbb R$ need not have a bounded derivative. There have been some attempts at giving ...
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0answers
34 views

Inverse image of $\tan(x)$.

Let $f(x) = \tan(x)$. What is $f^{-1}[\mathbb{R}^+]$? (i.e. what is the inverse image of the positive real numbers under $f$?) Would it just be the positive values of the domain of $\tan(x)$, i.e. ...
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1answer
27 views

Find two convex functions bounded below by $0$

I'm trying to find two convex functions $f$ and $g$ continuously differentiable in the interval $[0,1]$ such that $f'(x) = -g'(1-x)$ $\forall h > 0:f'(x) < f'(x+h)$