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

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0
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2answers
21 views

How to prove functions are odd and even

Show that any function f on [-a,a] where a is a positive constant, can be written as the sum of an even and an odd function?
0
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0answers
16 views

Unbounded Function

I am trying to find vales of $a >0$ s.t the function is unbounded on $[0,1]$ $f_a(x)= \begin{cases} x^{a-2}(ax\sin(1/x)-\cos(1/x)), & x\neq 0 \\ 0, & x =0 \end{cases}$
0
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0answers
10 views

Notational difference functions and mappings, talking sets and classes

The closest question to mine I could find was Is there any difference between mapping and function? I have something different in mind however. I use to use the term "function" when I'm talking about ...
7
votes
3answers
46 views

Function composition: $f^{653}(56)=?$

Let $f(x) = \frac1{(1-x)}$. Define the function $f^r$ to be $f^r(x) = f(f(f(...f(f(x)))))$. Find $f^{653}(56)$. What I've done: I started with r=1,2,3 and noticed the following pattern: $$f^r(x)= ...
0
votes
1answer
10 views

Linearization of a function in the point 0, 0

The linearization of the function $ f(x, y) = 1 + 2(x + 1) + 3(y + 1) + 4x^2 + 5y^2 $ in the point (0, 0) is given by: $ L(x, y) = 6 + 2x + 3y $ I know this is true, but how does one come to this ...
1
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3answers
17 views

For which of the following functions the $\sum_{x\in S(f)}\frac{1}{x}$ converges?

For real valued function $f$ define $$S(f)=\{x:x>0,f(x)=x\}$$ For which of the following functions the $\sum_{x\in S(f)}\frac{1}{x}$ converges? $\tan x,\tan^2x,\tan{\sqrt{x}},\sqrt{\tan x},\tan ...
0
votes
1answer
16 views

Composite Functions

$f(x)= \dfrac{1}{10x+17}+13$ $g(x)= \dfrac{1}{9x-6}$ I need to find $f(g(x)).$ How do I do this? I keep on getting it wrong. The correct answer is $\dfrac{1998x-1202}{153x-92}$. But I am unsure how ...
0
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1answer
14 views

A question regarding real valued function

I have a question regarding real-valued function: Which of the following cannot possibly be the rule of any real-valued function? A) $y=\sqrt{x-1}$ B) $y=\sqrt{x-1}+\sqrt[3]{2+x}$ C) ...
1
vote
1answer
10 views

Finding inverses of two functions and their compositions to solve for unknown.

$$f(x) = 23x + 27,\;\; g(x) = 12x - d$$ I've found $f^{-1}(x),$ and $\,g^{-1}(x)$, but I don't know how to solve for $d$, given $$f^{-1}(g^{-1}(x)) = g^{-1}(f^{-1}(x)).$$ How do I do this please?
1
vote
2answers
19 views

Which of the following is constant?

If $f,g$ are continuous real valued functions such that $f\circ g$ is constant then which of the following must be constant? $$f,g,g\circ f$$ I think when $f\circ g$ is constant then at least one of ...
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0answers
9 views

composition of Riemann integrable functions.

I have two functions: f:[a,b]->R and g:[c,d]->R where a My question is if it follows that g o f (the composite function of f,g) is Riemann integrable as well?
0
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3answers
43 views

Problem about a bijective map from $\mathbb R^2 \rightarrow (0,1)$ [on hold]

Does there exist a bijective map from $\mathbb R^2 \rightarrow (0,1)$? What will be the mapping?
1
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0answers
25 views

Prove whether a particular function is concave

Given the following equation: $$V(w) = - \frac{\alpha}{2} \left[ y_1(w) + y_2(w) + \int _{-\infty}^{+\infty} \vert y_1(w) - y_2(w) - x\vert f_{T1}(x)dx\right] \\- \beta \int _{w - y_1(w)} ^{+\infty} ...
2
votes
3answers
177 views

Homeomorphism(topological spaces) version of Cantor–Bernstein–Schroeder theorem

Let $A$ , $B$ be topological spaces such that there for some subset $D$ of $B$ there is a homeomorphism form $A$ to $D$ and for some subset $E$ of $A$ there is a homeomorphism form $B$ to $E$ ; then ...
0
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0answers
6 views

MR Function Question

I have a question about MR Function below: P = 1/Q^2 + 3Q + 1 Find the MR Function and Evaluate it at Q = 4 Are you guys able to elaborate? I've never done these types of questions before, am i ...
0
votes
1answer
28 views

Why have we made a function to be many to one and not one to many? [on hold]

We have allowed function to only relate many to one but not one to many. Why haven't we included sin(x) to be a function? Is it just for simplicity? Also, I've seen someone quote a function not even ...
2
votes
0answers
20 views

Real Analysis Differential functions

I am currently working through an exercise set and I am a little stuck on the following question: For $a > 0$, define a function $f_a(x)= \begin{cases} x^a \sin(1/x), &x \ne 0\\0, &x=0 ...
2
votes
1answer
51 views

How to find the function $f$ that satisfies $f(x, y) = f(x^{-1}, y^{-1})^{-1}$ and $f(x, y)$ is $\approx$ $average(x, y)$?

Fist of all, I'm a programmer, not a mathematician, and I'm sorry for my non native English. And I'm sorry if the question is not appropriate, it is my first time here. Or if the question has no ...
1
vote
1answer
26 views

Lyapunov exponent for simple functions

Context: We know that $\cos(x)$ if taken recursively on itself, converges to the Dottie number, which is the function's stable fixed point then. On the other hand, for a function like $f(x)=3x$, ...
-3
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0answers
11 views

“For each of the three surfaces, what is the stopping distance for a car traveling at 80km/h?” [on hold]

"The stopping distance of a car on dry asphalt can be modeled by the function $d(s) = 0.006s^2$, where $d(s)$ is the stopping distance, in meters, and $s$ is the speed of the car in kilometers per ...
1
vote
1answer
17 views

Drawing toral automorphisms

How do we set about drawing a toral automorphism as in figure 5.1 in the picture above. How do we know where the points highlighted in yellow are? What happens if the eigenvalue (Im guessing some ...
0
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3answers
23 views

Limit of a function w

If $f(x, y)$ is a continus function, defined in whole $\mathbb R^2$, then the limit $$\lim_{(x, y)\rightarrow(2,2)}f(x, y)(x-1)(y-2) $$ The solution is $0$, but how? A very elaborative explanation ...
4
votes
2answers
30 views

Making a piecewise function continuous and differentiable at point

Problem: Let $f(x) = \left\{ \begin{array}{lr} \frac{\arctan(x)}{(1+x)^2} & : x \geq 0\\ Ae^x + B & : x < 0 \end{array} \right. $ Find $A$ and $B$ such that the function is ...
4
votes
6answers
65 views

Domain of $\frac{1}{\frac{1}{x}}$

Let $f(x)=\frac{1}{x}$, then we have $f^{-1}(x)=\frac{1}{x}$. So $f(f^{-1})=\frac{1}{\frac{1}{x}}=x$. My question is, what is the domain of $f(f^{-1})(x)$? is it everything? or everything but zero? ...
1
vote
3answers
18 views

Intersection of inverse images

Given $A$ and $B$ is the subset of $C$ and $f:C\mapsto D$, $$f(A\cap B)\subseteq f(A) \cap f(B)$$ and the equality holds if the function is injective. But why for the inverse, suppose that $E$ and ...
0
votes
1answer
15 views

Integral of a normal function multiplied by heaviside and delta functions

$\int_{-\infty}^{\infty} e^{2t}u(\tau - t)t^{2}\delta(t)dt$ Hi! How would I go about computing this integral? I understand I can change one of the integration limits and eliminate the heaviside ...
0
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0answers
15 views

Questions about functions, their domains and codomains.

I am playing around with equations about functions in general and have some questions. Question 1 If I have some functions $f,g\colon X^2 \rightarrow Y$ such that $f(x,y) = g(x,c)/g(y,c)$ then can ...
0
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1answer
12 views

Composing Piecewise Functions

I was wondering how to compose piecewise functions. On a practice exam I was reading, a question asks what F(F(x)) will look like if F(x)= 2x if x<1/2 and = 2-2x if x>=1/2. Would I just ...
0
votes
1answer
39 views

How to calculate “general” integral $\int\limits_{a}^{b}f(x)^2dx$?

How to calculate "general" integral: $\int\limits_{a}^{b}f(x)^2dx$?
0
votes
1answer
37 views

What is this integration “method” name?

I see that people often write this equality: $$\int\limits_a^bf(x)\,\mathrm dx=\int\limits_{f(a)}^{f(b)}f(x)\,\mathrm df(x)$$ when dealing with functins in general, that is when something is trying ...
-3
votes
1answer
24 views

Arccos function [on hold]

How to look for condition (limits of x): Or, how to get domain limits for $-1 < = \frac{( x-1)} {(2 x +7) } < = 1 $ for using in -1 <= argument <= 1 ( to get real values of ...
1
vote
1answer
24 views

Inequality from a property of convexity

Let $f:\mathbb{R}\to\mathbb{R}$ be a convex function. How can I prove that for each $x$, there is $c$ such that $f(x)+c(y-x)\leq f(y)$ for all $y$? One of the difficulties to solve is $f$ does not ...
0
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0answers
24 views

Understanding integration and substitution

I'm an undergraduate student in EE. I often see that when talking about voltages, curents ... being expressed like functions of some independed variable (time) and when calculating integrals people ...
0
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1answer
26 views

Empty function, what is it?

I meet with term 'empty function' from time to time. It's high time to understand its nature. What is field( set of arguments) and what is image? ( set of value)?
0
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0answers
9 views

general inverse of multivariable function

Given $m<n$. Let $f:N\subset\mathbb R^m\mapsto \mathbb R^n$ be a differentiable function. I am looking for the condition(s) such that we can find a function $g:im(f(N))\mapsto \mathbb R^m$ ...
1
vote
1answer
33 views

Who is growing faster?

I am trying to prove that $\lim_{n\to \infty} { 2^{n^2} \over n!} = \infty$. I can't use l'Hôpital's rule (or I dnon't know how) and I don't recall any other method which could help me. It also isn't ...
0
votes
2answers
27 views

Limit as x approaches 0 from the left: $\lim_{x \to 0^{-}} \sin^{-1}\left({\frac{1}{2+e^\frac{1}{x}}}\right)$

Help me find the limit as x approaches 0 from the left: $$\lim_{x \to 0^{-}} \sin^{-1}\left({\frac{1}{2+e^\frac{1}{x}}}\right)$$ Thanks,
1
vote
1answer
11 views

Can the induced function of non-zero $f \in R[X]$ be zero, when $R$ is an infinite non-integral domain?

Let $R$ be an infinite commutative ring with $1$ which is not an integral domain. Is it possible to have a non-zero $f\in R[X]$ such that the induced map $\bar{f}: R \to R$ is zero? Please give ...
0
votes
1answer
33 views

Subspaces and annihilators

I am trying to show this question. My understanding of annihilators is that for a vector space $V$ over $K$, with $S$ being a subset, the annihilator of $S$ is the subspace $S^0$ of linear functions ...
1
vote
4answers
57 views

Find $\,\lim_{x\to 1}\frac{\sqrt[n]{x}-1}{\sqrt[m]{x}-1}$

How do I calculate $\lim_{x\to 1}\frac{\sqrt[n]{x}-1}{\sqrt[m]{x}-1}$. Please help me. Thanks!
-3
votes
0answers
17 views

Is these functions periodic ones? [on hold]

Determine whether the function is periodic. If it is periodic, find the smallest (fundamental ) period $f(x)=cos(2x) + 3sin(\pi x)$ $f(x)=sin(2x)-cos(5x)$ thanks in advance
0
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0answers
38 views

Rewriting a function to speed up computation

the purpose of this question is to find the fastest way to calculate a function on a (x,y,z) grid using a programming language. It is more related to maths however, as it concerns recasting a function ...
4
votes
2answers
125 views

What is the opposite category of $Set$?

In $Set$ the initial object is the empty set, and it has an unique morphism to each other object, namely $f=\emptyset$. However I find it difficult to think about the category ${Set}^{op}$, is there ...
-1
votes
1answer
11 views

Number of different functions

Suppose that $A$ has exactly $m$ elements and $B$ has exactly $n$ elements. How many different functions are there from $A$ to $B$? The answer is given by $n^m$ but i don't know how to get that And ...
1
vote
2answers
32 views

Unbounded function on compact interval?

So what are some unbounded function on compact interval, if there is any? Also, is the function $f:[0,\infty) \to \mathbb R$, $f(x)=x$ continuous?
2
votes
2answers
22 views

Does there exist a bijective mapping of an open interval with the corresponding closed interval having only finitely many points of discontinuity?

Given $a<b$, is there a bijection $f \colon [a,b] \rightarrow (a,b)$ such that $f$ be continuous except at finitely many points only? I know that there does exist a bijection of $[a,b]$ with ...
0
votes
0answers
41 views

Prove probing to be permutation

So I have been taught that probe sequences (h(k, 0), h(k, 1), ... , h(k,m - 1)) are meant to be a permutation (0, 1, ... ,m - 1), but how does one prove that? I was asked this question in an ...
0
votes
2answers
29 views

Find $\lim_{x\to\infty} \frac{e^{2x}-1}{e^{2x}+1}$ and $\lim_{x\to-\infty} \frac{e^{2x}-1}{e^{2x}+1}$

How do I calculate $\displaystyle \lim_{x\to\infty} \frac{e^{2x}-1}{e^{2x}+1} \ , \ \lim_{x\to-\infty}\frac{e^{2x}-1}{e^{2x}+1}$. Please help me. Thanks!
1
vote
0answers
19 views

Is a sine wave plus the sum of its odd harmonics symmetrical around the x axis at half the period of the fundamental?

I have a function such that $$x(t)=A_1 \sin(2 \pi f t+\phi_1)+A_2 \sin(2 \pi (3f) t+\phi_2)+...+ A_n \sin(2 \pi ((2n+1)f) t+\phi_n)$$ Is such a function symmetric around the point that is half ...
-4
votes
1answer
33 views

Riemann hypothesis: An query about the primes [on hold]

How can we describe the Riemann hypothesis easily? What is the connection between the Riemann hypothesis and prime numbers?