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

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0
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3answers
35 views

How to solve this problem on absolute value function?

If $a,b\in \mathbb R$ and be distinct numbers satisfying $$|a-1|+|b-1|=|a|+|b|=|a+1|+|b+1|$$ then the minimum value of $|a-b|$ is ? ($|...|$ represents absolute value) I tried solving the ...
0
votes
0answers
35 views

How much can we compress a sequence of bits?

Suppose that we have a sequence of bits defined for each natural less than $n$. For example, if we have 3 bits (either 0 or 1), we can represent the sequence as a function of $x < 4$ in the ...
0
votes
2answers
41 views

If $f(x)=3x-4$, $f\circ g^{-1}(x)=5x+7$..

If $f(x)=3x-4$, $f\circ g^{-1}(x)=5x+7$ and $g(x)=5f(x)+7$, find the value of $x$ What is the difference between these two solutions. If there are no any differences then why do the answers not ...
1
vote
2answers
28 views

Help with proof that that $f(x)=\cos x\,\,\,[0,\pi]\rightarrow[-1,1] $ is one to one and onto

Is there a way to prove that $f\,:\,[0,\pi]\rightarrow[-1,1]\,\,\,\,f(x)=\cos x$ is one to one and onto ? I know that for $[0,\pi]$ cosine is strictly increasing function therefore it has to be one ...
0
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2answers
47 views

Axiom of Choice and LEFT inverse [duplicate]

I am aware of why the Axiom of Choice is equivalent to the the statement that every surjection splits. However, I don't see why we don't also need AC to show that every injection splits. In ...
0
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1answer
24 views

Minima of two convex functions that are “close” to each other

Consider two convex functions $f_1 : \mathbb{R}^n \to \mathbb{R}$ and $f_2 : \mathbb{R}^n \to \mathbb{R}$ such that $\hspace{2cm} |f_1(x) - f_2(x)| \leq \epsilon \hspace{2cm} \forall x \in ...
0
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0answers
27 views

It is possible to prove that the inverse of every periodic function is a multivalued function?

As I state in the title, I wonder if is possibile to prove that the inverse of every periodic function is a multivalued function. First of all I can't found a counterexample for the statement, and ...
2
votes
1answer
57 views

How to graph $\frac{(x+3)}{(x+1)}$?

I'm not looking for plotting values. I mean something which can be done without it. For example, for the function $f(x) = mx + b$, we can use the slope $m$ and the y-intercept $b$ to easily graph the ...
0
votes
1answer
45 views

Prove continuity of piecewise function using epsilon-delta

Suppose we have a function $\phi$ so that $$\phi (x)=\cases{f(x) & \text{ if } x\le 0\\ g(x)& \text{ if } x>0.}$$ where $f$ is continuous on $(-\infty,0]$ and $g$ is continuous on ...
1
vote
0answers
50 views

Properties about a function on $R$ [duplicate]

I have no clue about the following (supposedly simple) question from Mathematical Analysis: Let $f: [a,b] \to \mathbb{R}$ be differentiable on $[a,b]$. Suppose that $f'(a)=f'(b)=0$, and that $f''$ ...
1
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2answers
29 views

if f is a function with domain $\mathbb{R}$ that can be written $f = E + O$ where E is an even function…

if f is a function with domain $\mathbb{R}$ that can be written $f = E + O$ where E is an even function and O an odd function, prove that writing $f$ in this way is unique. Sol'n: A solution that I ...
2
votes
3answers
31 views

Proof continuity of a function with epsilon-delta

I quickly need help with a problem that seems to be fairly easy but I can't really do the final step: Proof that the function $\frac{x-1}{x²+1}$ is continuus in $x = -1$ using the ...
0
votes
1answer
15 views

Find Monotony of $(fx)$ from the equation of $f'(x)$

I have $f'(x)$ drawn on graph and I've deduced that its linear equation of the form $f'(x)=-3x+3$. Question is: $f(x)$ is increasing when $x \in $ ...... ? image of the question
2
votes
2answers
84 views

Inverse of function $f(x)=x^3+3x+1$

If $f:R \to R, f(x)=x^3+3x+1$ then find the inverse of $f(x)$. $f(x)$ is one to one (as it is increasing function for $x \in R$) and onto as well (Range is same as co-domain) but I don't know how to ...
0
votes
1answer
17 views

continuity of function series

So here it is: $$\sum_{n=1}^\infty \frac{\sin(\frac{1}{nx^2})}{1+(x-1)\ln^4(xn)}$$ $$x \in (1,\infty)$$ My task is to prove its continuity if possible. My lead was to try proving it through ...
1
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0answers
36 views

discrete random variable with uniformely distributed random variable

I hope you can help me because I have no clue where to start: Let X be a discrete random variable with $ p_k=P_X[X=x_k]=p(x_k) $for all $1\le k\le N$ for $N\in \Bbb N$ and distribuition function: ...
1
vote
2answers
36 views

Dividing an integer into a fixed number of integers

What is the formula for dividing an integer into a fixed number of integers where the greatest distance between consecutive integers is 1. Dividing 10 into 4 integers we can get: ...
0
votes
0answers
18 views

Function for asymmetrical pyramid with a rectangular base

I would like to derive a single function for a pyramid (corners can be rounded) that has it's tip always centered on x=0,y=0 and the size and position of the base - dynamic, depending on the ...
3
votes
5answers
84 views

Is the function $f(x) = x^2 - 2x$ one-to-one?

The answer given is that it's not because it's a parabola and hence, would fail the horizontal line test, i.e., two values of $x$ will have the same value of $f(x)$. However, how can we prove the ...
0
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2answers
41 views

Is $f''(x)=0$ sufficient for inflection point?

I'm a bit confused about $n$th derivative test.Is $f''(x)=0$ at a point sufficient to prove it is inflection point or not ?Or we need to check further if any higher odd derivative is $0$? And when ...
2
votes
1answer
61 views

Function inversion (analytical)

Can $t(x)$ be found from: $$A \, t + B\ln\frac{1-t}{t}=x \; ?$$ Here, $A>0, \; B < 0$ and $0 \lt t \lt 1$. The $t(x)$ should be given in analytical form (even if you use, say, Lambert's W - ...
1
vote
1answer
58 views

Let $f(x) =x+\tan^3x $ and $g(x) =f^{-1}(x)$ then find the value of …

Problem : Let $f(x) = x + \tan^3 x $ and $g(x) =f^{-1}(x)$ then find the value of $56g'(\frac{\pi}{4}+1)$ My approach : $f'(x) = 1+3\tan^2x \sec^2x$ $f'(\frac{\pi}{4})=7$ now how to find ...
2
votes
1answer
26 views

Let $f: [0, \infty ) \to \mathbb{R}$ be a continuous and strictly increasing function such that $f^4(x) =\int^x_0 t^2f^3(t)\,dt$ for all $x > 0$

Problem : Let $f: [0, \infty ) \to \mathbb{R}$ be a continuous and strictly increasing function such that $f^4(x) =\int^x_0 t^2f^3(t)\,dt$ for all $x > 0$. Find the area enclosed by $y = f(x)$, ...
-3
votes
1answer
34 views

Show that this function is strictly CONCAVE [closed]

Please help me show that $f(w)$ is strictly concave in $w$: $f(w)=\sum_{j=1}^N P_j (w)\cdot u_j $ over $w\in [0,\infty)$ where $P_j (w)=\sqrt{w}\int _{-\infty}^{\infty}\Pi_{k\neq ...
0
votes
2answers
48 views

If $f$ is a morphism then $x\leq f(x)$

$\mathcal{A}=(A,\leq_A)$ with $\leq_A$ a well order over $A$ and $f: \mathcal{A}\to \mathcal{A}$ is a morphism. Then $x\leq f(x) \;\forall x \in \mathcal{A}.$ I think the proof should be obvious ...
0
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0answers
16 views

Proving that a function must be constant using Complex Analysis [duplicate]

Fix nonzero ω1,ω2 ∈R. Suppose that f is an entire function which satisfies $$f(z + ω1) = f(z + iω2) = f(z)$$ for all $z ∈\Bbb C$. Prove that f must be constant. My first and immediate thought upon ...
0
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0answers
17 views

methods function approximation

My goal is to modelize the temperature of a motor: I want to be able to calculate the temperature of a motor at a given moment, with a given torque and a given speed, in a constant environment. For ...
0
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0answers
11 views

How to regularize an irregular curve?

I have a curve that has a couple of discontinuities and changes the sign ofd the 2nd derivative. I want to fit this curve with a second curve, that have to be continuous and with positive 2nd ...
0
votes
1answer
18 views

What are the domain and name of a function that takes a vector of arbitrary length as argument?

Is there a name for a function that takes any vector (or list) as argument? And what is its domain? For example, if $v$ is a vector of arbitrary length, what is the domain of functions such as: ...
3
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0answers
56 views

Find the polynomial $p(x)$

A polynomial $p(x)$ gives a remainder of $1$ when divided by $x^{100}$ and a remainder of $2$ when divided by $(x-2)^3$. Evaluate $p(x)$. By the Remainder Theorem, $p(x)$ can be written as ...
1
vote
1answer
13 views

How do I plot this graph in octave?

Trying to plot this graph in octave, do anyone know how to do it? I do not know how to do it when it is equal to 0. $r^{3}(1 + \frac{0.00822}{101325 * r}) - r(\frac{6 \pi * (16.7 * 10^{-6}) * (1.6666 ...
0
votes
1answer
49 views

Regarding absolute continuity of some function

$f (y) $ is continuous function of y. $\int_{-\infty}^\infty |f(y)||(x-y)|^2dy$ is finite for all x Given $h(x)= \int_{-\infty}^\infty f(y)(x-y)^2dy=\int_{-\infty}^\infty f(y+x)(y)^2dy$ is $h(x)$ ...
0
votes
3answers
43 views

Is there a way to combine functions so that you combine their derivatives?

Suppose $y,z$ are functions. What manipulation: "$?$" to the functions would yield the following? (if any) $$y?z=y\cdot z\\~\\ \frac {d(y?z)}{dx}=\frac{dy}{dx}\cdot\frac{dz}{dx}\\~\\ \frac ...
0
votes
2answers
33 views

A function that satisfies the $n$-th derivative where $x=0$ is $\frac{1}{n}$ [closed]

Is there a function that satisfies $f^{(n)}(0)=\frac{1}{n}$ for every positive integer $n$?
4
votes
3answers
84 views

What is the reason for naming a function odd or even [duplicate]

We say that a function is called odd if $$f(-x)=-f(x)\\ (1)$$ and a function is called even if $$f(-x)=f(x)\\\\\\ (2)$$ But why do we call them odd and even. It feels a very peculiar choice of ...
0
votes
1answer
21 views

Prove: If f and g are two uniformly continuous functions in I, then $\alpha f+\beta g$ is also uniformly continuous in I

Prove: If f and g are two uniformly continuous functions in I, then $\alpha f+\beta g$ is also uniformly continuous in I Where $\alpha , \beta \in R$ and I is a section that can be closed or not. ...
1
vote
2answers
40 views

Finding a formula for the number of functions

Let $P_{k}$ denote the set of all subsets of $\{1,2,,\ldots,k\}$. Prove that the number of functions $f$ from $P_{k}$ to $\{1,2,\ldots,n\}$ such that $f(A\cup B) = \max(f(A), f(B))$ is $1^k +2^k + ...
0
votes
1answer
60 views

Recurrence relation problems

For some math homework (that was already due but I really want to understand the content) I was asked the following question, How should I go about answering this? I'm new to recurrence relations and ...
0
votes
2answers
62 views

Greatest Integer Function $ [x^2] $ : Riemann Integration Question

How to show $$ [x^2] $$ is Riemann Integrable in [0,2] ? I will explain how I proceeded my doubt is with greatest integer function part, after splitting into 3 intervals $$[0,1],[{1,\sqrt 2 }], ...
0
votes
1answer
73 views

Is 4.99999… exactly equal to 5? [duplicate]

I'm a student of 10th std. Recently our teacher asked a Question that "Is 4.999...equal to 5 or not?" Everyone said that is isn't equal or it is approximately equal. Teacher too agreed to that. But ...
1
vote
3answers
41 views

How to graph $x^2 -4x$?

I know about transformations and how to graph a function like $f(x) = x^2 - 2$. We just shift the graph 2 units down. But in this case, there's an $-4x$ in which the $x$ complicated everything for me. ...
1
vote
2answers
21 views

Using Symbolic Logic to keep track of domain and codomain

I'm working on Spivak's Calculus, and trying to throw Symbolic Logic at it to make its exercises a little more formal. Chapter 3, Problem 1 VII is: Let $f(x) = \frac{1}{1+x}$. For which numbers is ...
-2
votes
2answers
41 views

Solve $5\sin x = - 2$ where $0 \le x \le 2\pi$ [closed]

Solve $5\sin x = - 2$ where $0 \le x \le 2\pi$.
1
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0answers
22 views

Stating the domain of a function using set notation

I'm trying to write the domain of $f(x+y) = \frac{1}{1+ x + y}$ given $f(x) = \frac{1}{1+ x}$ using set notation. I'm thinking first we'd have something like $Domain[ f(x) ] = \mathbb{R} \setminus ...
2
votes
3answers
416 views

How to calculate this integral when the function is unknown?

Let $f :[0,1]\to \mathbb{R}$ with $f'(x) = \sqrt{1+f^2(x)}$ for all $x \in [0,1]$. If $f(0) + f(1) = 0,$ calculate the integral $$I=\int_{0}^{1}f(x)dx$$ Any help would be appreciated. Thanks.
2
votes
2answers
63 views

sum of a function series

what would be the first step to determine sum of $\sum\limits_{n=1}^{\infty}ne^{-n^2/4x}.$ I think I should try putting $y=e^{-1/4x}$. Then $y$ changes from $0$ to $1$ and I get ...
0
votes
0answers
23 views

On proving that the surjectivity of a function is implied by the existence of a right inverse.

As I've mentioned in a previous question, the definition of a surjective function has been giving me some trouble however, I think that along with the answers to that question, this should resolve ...
-2
votes
2answers
45 views

Determine the solutions to the equation $2\sin x - \sin^2x = \cos^2x$ [closed]

Determine the solutions of the equation: $2\sin x - \sin^2x = \cos^2x$ for $0 ≤ x ≤ 2π$
0
votes
0answers
22 views

What value of 'a' will be the function is convex, concave or not either?

$$f(x,y) = -6x^2 + (2a+4)xy - y^2 + 4ay$$ The solution has to be : $$-2-\text{gyök}(6) \leq a \leq -2 + \text{gyök}(6)$$ I tried to define the derivation of the function accordance with $x$ , and ...
2
votes
5answers
59 views

Prove $\frac{2\cos x}{\cos 2x + 1 }= \sec x$

Prove that $\dfrac{2\cos x}{\cos 2x + 1 }= \sec x$. So far I have: $\dfrac{2\cos x}{\cos 2x + 1 }= \dfrac 1 {\cos x}$ Where do I go from here?