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

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0
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1answer
16 views

Function without any digital help.

Ive come into some trouble answering this function: $$h(x)=-0.05(x^2)+x+2.20 $$ $x$ and $h(x)$ are both shown in meters. I need to solve this function with $h(0)$. would be nice is someone could ...
0
votes
2answers
100 views

If $f^{-1}(x)=\frac{1}{f(x)}$ then find $f(1)$

For $a>1$ we have: $f:[\frac{1}{a},a]\to [\frac{1}{a},a]$ be a bijective function. Suppose $f^{-1}(x)=\frac{1}{f(x)}$ for all $x \in [\frac{1}{a},a]$ then find $f(1)$. Could someone give me ...
1
vote
1answer
35 views

Find zeros of a function or at least say things about their location?

Let $a>0$ be a fixed parameter. I would like to find the (I think there are only two) $x\in \mathbb{R}$ such that $$(x-a)e^{-\frac{1}{2}(x-a)^2} = (x+a)e^{-\frac{1}{2}(x+a)^2}.$$ I know this might ...
-1
votes
0answers
47 views

How to find f(x) from some unknown curve

Hi i am studying civil engineering and i have one interesting problem. For example: My task was to calculate the area of gravel located on the beam.So tell me exactly what information i need to find ...
0
votes
1answer
50 views

Function and inequality problem

Let $f$ an ascending and convex function on $(0,+\infty)$. I must to prove that: $$ f(\vert \sin(x) \vert +3) -f(\vert \sin(x) \vert)< f(x+3)-f(x) $$ I know that a solution of that is to ...
4
votes
1answer
40 views

An always increasing function

Suppose I wanted a function $f(x)$ such that the following properties are had. $f(x)$ maps $\mathbb{R}\to\mathbb{R}$. $f(a)>f(b)$ if $a>b$. The function may or may not be continuous, but it ...
2
votes
1answer
23 views

Concave up theorem for $f:A \to\mathbb R, A \subseteq \mathbb R $ - True or false?

I am a 1st year-2nd semester student of the department of Applied Mathematics @ NTUA. Some days ago, I had a really interesting conversation with one of my old school mathematics teacher about a ...
0
votes
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
40 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
votes
2answers
45 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
votes
1answer
23 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
votes
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
41 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
vote
2answers
28 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
30 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
83 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
vote
0answers
35 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
votes
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
58 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
votes
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
votes
0answers
16 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
votes
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
votes
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
42 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
61 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 }], ...
-1
votes
1answer
71 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. ...
0
votes
0answers
7 views

Pseudo-code for uniform kernel shape/function [closed]

I am trying to create a "kernel"-type function to be used as weights while averaging a set of numbers. I want to generate a variable using the formulas for the following kernel shapes: gaussian, ...
-3
votes
0answers
50 views

Solve $5 \sin(x) = −2$ where $0 ≤ x ≤ 2π$ [closed]

Solve $5\sin{x} = − 2$ where $0 ≤ x ≤ 2π$. So far I have: $\sin{x} = -\frac{2}{5}$ How would I get the x answers in fractions of $π$ from here?