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

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21 views

continuity of linear functional on family of functions

If $A$ : $C[a,b]\rightarrow \mathbb{R}$ is a continuous linear functional, then $ t\mapsto A(f_{t})$ is a continuous function on $\mathbb{R}$. where \begin{align} f_t(x)= \left\{ \begin{array}{lr} ...
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2answers
35 views

If $f\colon X\to Y$ is injective there is a $g\colon Y\to X$ such that $g\circ f=Id_X$

How to prove that if $f\colon X\to Y$ is injective there is a $g\colon Y\to X$ such that $g\circ f=\operatorname{id}_X$. I know that it is an if and only if, but I have already proved the reciprocal. ...
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2answers
56 views

A difficult problem on functions

I've been trying to solve the following problem but can't wrap my head around it. Let $x$, $f(x)$, $a$, $b$ be positive integers. Furthermore, if $a > b$, then $f(a) > f(b)$. Now, if $f(f(x)) = ...
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2answers
58 views

Derivative of function $f(x) = \sqrt{2x}+ \sqrt{2/x}$

The derivative of function $$f(x) = \sqrt{2x}+ \sqrt{2/x}$$ Here's what I did, $$f(x) = \sqrt{2x}+ \sqrt{2/x} \\ = (2x)^{1\over2} + ({2\over x})^{1 \over 2}\\\\$$ $$f'(x)={1\over 2}(2x)^{-{1\over ...
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1answer
22 views

A continuity result

Suppose (i) $f:R^n_+\to R$ and (ii) $f(x)=f(\alpha x)$, for all $\alpha>0$ and (iii) For any $x,y\in R^n_+$, if $x_n\to x^*$, $y_n\to x^{*}$, we have $\lim f(x_n)=\lim f(y_n)$. Could I claim ...
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0answers
21 views

Example of a continuous non-lipschitz function with domain $[0,1]$ and co-domain $\mathbb R$

I would like an example of a function which is continuous with domain $[0,1]$ but is not Lipschitz continuous. Is this possible? I know a continuous function with domain $[0,1]$ is uniformly ...
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1answer
33 views

Function and Derivatives

If $$F(x)= (x-1)^{20} - (x-2)^{30} \cdot(x-3)^{40}$$ The number of real roots of $F''(x)=0$ are? $F''(x)$ - Second Derivative of $F(x)$. I have worked it out by simply differentiating it and then ...
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1answer
29 views

If a function $f$ is invertible can I say that $f^{-1}$ is also one to one and onto?

If we have a function $f$ that is both one-to-one and onto (so it's invertible). Its inverse function $f^{-1}$ is also one-to-one and onto? If this is not true can someone please explain it to me or ...
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2answers
56 views

How to show that $f_{n+1} \leq f_n$?

Let $$f_n(t) = \frac{e^{nt}-e^{-nt}}{e^{nT}-e^{-nT}}$$ be defined on $t \in [0,T]$. Can someone help me, I need to prove that $f_{n+1}(t) \leq f_n(t)$ for every $t$. I tried taking ratios and/or ...
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2answers
30 views

Question about functions and topology

This is a very general question, but one that I have been struggling with. If we say that a function from a topological space X to a topological space Y is ONTO, then does that mean that for each open ...
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0answers
18 views

How to rotate a 2nd derivative gaussian function?

I have a 2nd gaussian derivative in y and a normal gaussian in x, which results in the function: $$ f\left ( x,y \right ) =\frac{- \exp ^{-\left (\frac{x^{2} +y^{2}}{2\cdot \sigma^{2} } \right ...
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1answer
24 views

“Positively homogeneous of degree zero”

I am trying to understand a statement in an economics paper (and this paper is unfortunately quite sloppily written). Let $A$ be a finite set. Let $S$ be the set of real-valued functions on $A$, i.e. ...
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0answers
38 views

All primes in the form 4x + 1 can be written as a sum of two squares. [duplicate]

Because all primes other than 2 are odd, one of the two perfect squares must be odd, with the other being even. Is there any way to prove the statement, or is it just an observation?
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1answer
10 views

Gaussian blur filter weight calculation formula is not clear

Here you can see Gaussian blur filter weight calculation formula: ...
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1answer
46 views

Question on construction of entire functions

Suppose that $x_i$ and $y_i$ are sequences in $\mathbb{C}$. Can you construct a non constant entire function such that $f(x_i)=y_i$? What happens if $x_i$ have an accumulation point? or what happens ...
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0answers
18 views

function notation: parameters of transformed functions

given the function $f(x)=x^6$, what are the parameters of the transformed function $g(x)=\frac{1}{2}\left(3-\frac{1}{2}x\right)^6-2$, what is the effect of each parameter on the graph of the original ...
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2answers
21 views

How to find the domain interval of an inverse function

I am taking an online calc 2 class and have come across something that I am struggling to understand how to do. Unfortunately, I am not teaching myself correctly and I have been unable to meet with ...
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6answers
122 views

How to prove $1+x \leq e^x~\forall x \in \mathbb{R}?$

How to prove $$1+x \leq e^x~\forall x \in \mathbb{R}$$ I'm stuck, I tried taking logs but didn't know how to proceed.
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0answers
34 views

Check my proof of a property of the greatest integer function?

Prove that $\forall n \in \mathbb{Z}, \lfloor x + n \rfloor = \lfloor x \rfloor + n $. Proof: Let $K = \{\ k\ |\ k\in\mathbb{Z},\ k \leq x+n\}$. Then, by definition, $$ \lfloor x + n \rfloor = j ...
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1answer
38 views

Is a maximum function concave?

A function such as $$\max (X, Y)$$ given $X\geq 0$, $Y \geq 0$. Because It would form an inverse L shape on the graph, and that looks like a concave function. However my teacher said that the function ...
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0answers
12 views

How can I know how a transient function is obtained?

In the following transient function, see if it can be obtained with r<1 or r>1. Image also here I have no idea how to do that! I know that z = r e^-jw, with r different than 1... do I need to ...
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1answer
63 views

Rudin's definition of derivative

Walter Rudin's Principle of Mathematical Analysis defines the derivative as follows in Definition 5.1: Let $f$ be defined (and real-valued) on $[a,b]$. For any $x \in [a,b]$ form the quotient ...
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1answer
25 views

Finding the Inverse of this function

Im trying to find the inverse of this function $$x \mapsto\frac {113^x - 1}{112}\def\comment#1{}\comment{(pow(113.0, x)-1.0)/112.0} $$ But it always turn up incorrect. Can someone point me in ...
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2answers
33 views

How to find the function for six step operation

I am trying to find a function for the following scenario: Rotating the red arrow will produce a nice sine wave as illustrated to the right of the hexagon. But I need to rotate the blue arrow, and ...
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1answer
28 views

Finding the inverse of a function in two variables

I have a function $f$ on the integers in $[-180,180)\times [-90,90)$ defined by $$f(y,x) = y + 360 x$$ I would like to find the inverse function. How can I do this?
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1answer
43 views

question involving remainder of complex function

The question says - Dividing $f(z)$ by $(z-i)$, we get remainder $i$ and dividing by $z+i$, we get remainder $1+i$. Find the remainder upon division of $f(z)$ by $z^2 + 1$ How do I go about ...
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1answer
25 views

Finding coefficients of a function, given a list of points on the function

Given $f(x) = ax^n + bx^{n-1} + ... + cx + d$, a list of points, and a specification of a tangent line (point $p_t$ and equation) find $a, b, ..., c, d$ s.t. $f(x)$ passes through each point the ...
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2answers
13 views

Notation for a function with multiple return values

I want to define a function $f$ whose domain is given by the set $V$ whose return value is a subset of $C$. Please correct me if I am wrong, I assume that $f : V \rightarrow C$ would mean that the ...
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0answers
32 views

Differential equation - fractions, circular answer?

Hi this might seem like a really stupid question but then hopefully someone can asnswer it quite easily :) I have function $P{_t}$$=(E{_t}$ $(P{_t}{_+}{_1}+$ $δ{_t}{_+}{_1}$$ )-γΩx$${^*})/$$(1+rf+ψ_t ...
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0answers
9 views

Abel transformable flat-top function

I am looking for an even function that has flat profile at $r=0$ and relatively flat to some value of $r$ and then drops down to nearly zero. It is similar to the graph below, where it drops around ...
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1answer
27 views

Mathematical formulation for equation in a loop

How can somebody write in a mathematical framework for a recursively defined function, for example: $ f = f(t,x,g(t,x)) $ and $ g = g(t,x) $ for i = 1:t_end $ ...
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1answer
29 views

If $f_n(x) = \int_0^x f_{n-1}(t) dt, x \geq 0 $, then $f_n(x) = \frac {1}{(n-1)!} \int_0^x f_0(t) (x-t)^{n-1}dt$.

If $f_0$ is continuous on $[0, \infty)$ and for all $n \in \mathbb N$, $f_n(x) = \int_0^x f_{n-1}(t) dt, x \geq 0 $, then $f_n(x) = \frac {1}{(n-1)!} \int_0^x f_0(t) (x-t)^{n-1}dt$. It can be easily ...
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2answers
61 views

$f : \mathbb R \to \mathbb R $ be a differentiable and $f'(x) $ is bounded function then $f$ is unbounded.

$f : \mathbb R \to \mathbb R $ be a differentiable function with $f(0) =0$. If for all $x \in \mathbb R , 1< f'(x) < 2$ then $f$ is unbounded. We know that when $f'(x) > 0$ then $f$ is ...
2
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1answer
9 views

Finding the function f(t) from it's graph

Here's what I have so far: $$f(t) = (-2(t+1)+1.5) \times (u(t+1)-u(t)) + (t-0.5) \times (u(t)-u(t-1)) + 0.5\cos(\pi t) \times (u(t-1)-u(t-3))$$ I found the majority of this function, but I'm not ...
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2answers
25 views

Getting 25% growth over last year sale

I want to calculate 25% growth over prior year's sale (by month). The way I am solving this currently is by multiplying (sale)*1.25; this works fine for sale values that are positive but for those ...
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0answers
14 views

PDF & CDF Differentiation &Integration

Let $F(.)$ be the cumulative distribution function of a $N(0, 1)$ random variable. Define a function $g(x)$ as $$g(x) = \int_{-\ln x}^{x^2}F(t + x) dt.$$ Evaluate $g'(1)$ in terms of $F(.)$, where ...
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2answers
23 views

Showing internal angles of a square are unaffected by a mapping

I recently had an exam in complex analysis, and I am slightly confused by one of the questions, so I'd appreciate any clarification: The mapping from the complex $z$ plane to the complex $w$ plane ...
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2answers
29 views

find the values of the constants where the function is continuous, but not differentiable.

For the following function, find the values of the constants $a$ and $b$ for which the function is continuous, but not differentiable. $f(x)=ax+b, x>0$ $f(x)=\sin x, x \leq 0$ I ve found out, ...
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2answers
42 views

Can ANY 2 or 3 dimensional shape be reversed engineered to give an equation (formula) for its shape??

Can ANY 2 or 3 dimensional shape be reversed engineered to give an equation (formula) for its shape? In other words given ANY 2 or 3 dimensional shape that ones draws on a graph can one reverse ...
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1answer
49 views

Find the interval(s) in which function is dercreasing.

For what values (or intervals) of 'a' it holds $a(a-1)x^{a-2}(x-2)+2a(x+1)^{a-1}<0$, where $x\ge2$. I tried to do it by first derivative test but it again gives almost same type expression which ...
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4answers
77 views

If $f(x)=8x^3+3x$ , $x\in\mathbb{R}$, how do I find $\lim_{x \to \infty}\frac {f^{-1}(8x)-f^{-1}(x)}{x^{1/3}}$?

Let $f: \mathbb R \to \mathbb R$ be defined as $f(x)=8x^3+3x$. Then $f$ is continuous , strictly increasing, and $\lim _{x\to \infty}f(x)=\infty , \lim_{x \to -\infty}f(x)=-\infty$ , so $f$ is ...
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0answers
37 views

Sketch the level curves of the function $f(x,y) = (x+y)^4$

My attempt: Let $$z = (x+y)^4 \iff \pm \sqrt[4]{z} = (x+y)$$ Since $z$ is a constant, $\sqrt[4]{z}$ will also be a constant. $$(x+y) = \sqrt[4]{z} = c \iff y = -x + c$$ So the contour plot will ...
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3answers
49 views

How can I check if a function is always $\ge 0$ for all values of its parameter $\ge 0$?

Let's say I have a function with with arbitrary coefficients and powers, something like $f(x) = 5x^2 - 7x^3 + x^4$. How can I check if this function is always $\ge 0\ \forall x \ge 0$? The procedure ...
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2answers
42 views

When I parameterize a function, what happens to its dimensionality? Specifically, in the context of the implicit function theorem.

Suppose I have some functions $f,g$ such that $$f:\Bbb{R} \mapsto \Bbb{R}^2$$ $$g:\Bbb{R}^2 \mapsto \Bbb{R}^n$$ My Question: For some $c \in \Bbb{R}$, is $g(f(c))$ a function of one variable? If ...
6
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6answers
82 views

Range of $\sqrt{x-1}$

Problem: Find the range of $f(x)=\sqrt{x-1}$ Now the problem I face is this: is the range $[0,\infty),$ or is it $(-\infty,\infty)$? $$$$I had learnt that $\sqrt{x^2} = \pm x$. However, on the ...
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0answers
13 views

How Do I Include a Scaled Variable into a Computation?

I'm trying to calculate the fall time of an object that has some gliding properties. Meaning it can convert velocity into lift at a small ratio. If I am given that lift is a linear function of ...
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0answers
30 views

Generalized Functional Equation

$ f\circ f\circ f(x) = f(3x) $ Ignoring trivial (constant) solutions, I am not sure what I can try as an initial guess. Also, how does this generalise? i.e. If $f^k(x) = f \circ f \circ f\circ ...
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1answer
78 views

f and g are bounded with domain of integers and target the real numbers . If f/g is bounded, then g/f is bounded.

I have come up with two bounded functions f = 1/x^2+1 and g = 1/x^2+2 and these tell me that g/f is also bounded. However, I am having trouble writing a proof or proving that g/f is not bounded by ...
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1answer
80 views

f and g are bounded . if 1/g is bounded, then f/g is bounded.

I would like some help understanding how to go about this question. I think that f/g is not bounded, but I cannot figure how to show that f/g is not bounded.
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1answer
22 views

is the function $f(x,y)=(x\sin(y),x\cos(y))$ on the given interval injective?

Consider $f:(0,\infty)\times (0,3\pi )\to \mathbb{R}^2$, given by $$f(x,y)=(x\sin(y),x\cos(y)).$$ Is $f$ injective? I have to find $(x,y)\in f:(0,\infty)\times (0,3\pi )$ such that $x\not= y$ but ...