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

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1
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2answers
50 views

Are there other functions of sets $f$ such that they have this property?

$f(A \cap B) = f(A) \cup f(B)$ This function is similar to the $\log_c$ function in that application of it onto a multiplication is equivalent to the summation of its applications. $\log_c(ab) = ...
1
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1answer
37 views

Equivalence class of functions with commutative diagram.

Let $S$, $T$ be sets, and $f,g: S \to T $ be function satisfying a condition that, there exist $\phi : S \to S, \rho : T \to T$, bijections, such that $f = \rho^{-1} \circ g \circ \phi$. Then we call ...
1
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1answer
23 views

How to prove $f$ is 1-strongly convex convex if and only if $f - \frac{1}{2}\|\cdot\|^2$ is convex?

I am trying to prove that a function $f:Z \mapsto \mathbb{R}$ is 1-strongly convex if and only if the function $f - \frac{1}{2}\|\cdot\|^2$ is convex. Assuming that $f$ is strongly convex, I have by ...
0
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0answers
18 views

sinz transformation of domain

Given the domain $$D = \{ x+iy \; \; | \; \; x < 0, y>0 \}$$ To where the function $\sin z$ transform D ? The answer is : ellipse where $ a=\cosh y$, $ b=\sin y $ and the domain of $u/v$ ...
0
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1answer
14 views

Clarification on an excerpt involving number of functions

Here is an excerpt from a book I am reading: Consider an agent that has to recognize letters of the alphabet. Suppose the agent observes a binary image, a $30×30$ grid of pixels, where each ...
3
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1answer
64 views

Create parameterizable map between $\mathbb{Z}$ and $\mathbb{Z}^2$

I would like to create a parameterizable map between $\mathbb{Z}$ and $\mathbb{Z}^2$. This map I'll call $M$ and the parameter I'll call $k$. $k \in \mathbb{Z}$ (but if there is a better space for ...
6
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5answers
118 views

What is the most used method for proving continuity for simple functions such as $f(x) = x^{1/3}$

In analysis we talked about a very general definition of continuity: $f:A \to B$ is continuous if $U \subset B$ is open, $f^{-1}(U) = V \subset A $ is open Quite elegant Another definition is if ...
1
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2answers
32 views

Let $f:[1,\infty)\to R$ be a monotonic and differentiable function and $f(1)=1.$If $N$ is the number of solutions of $f(f(x))=\frac{1}{x^2-2x+2}$.

Let $f:[1,\infty)\to R$ be a monotonic and differentiable function and $f(1)=1.$If $N$ is the number of solutions of $f(f(x))=\frac{1}{x^2-2x+2}$.Find $N.$ ...
7
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5answers
150 views

Existence of an injective continuous function $\Bbb R^2\to\Bbb R$?

Let's say $f(x,y)$ is a continuous function. $x$ and $y$ can be any real numbers. Can this function have one unique value for any two different pairs of variables? In other words can $f(a,b) \neq ...
2
votes
1answer
64 views

Set of Discontinuities for a function $f$

Take $f$ to be a function over the reals. I want to show that a set of discontinuities of the first kind for $f$ are countable. This is the discontinuity type at point $P \in \mathbb{R}$ where $lim_{x ...
3
votes
0answers
50 views

Same values for Gamma Function

I was thinking about the Gamma function, which for an integer positive argument is nothing but the factorial function. Using the integral representation, namely $$\Gamma[x] = \int_0^{+\infty}\ ...
4
votes
3answers
90 views

how to find all functions such that $f\left( x^{2} - y^{2} \right) = ( x - y )( f(x) + f(y) )$

Find all function $f:\mathbb R\to\mathbb R$ such that $f\left( x^{2} - y^{2} \right) = ( x - y )( f(x) + f(y) )$. My try: If $ x=y=0$ then $f(0)=0$ and if $x\leftarrow\frac{x+1}{2}$ and ...
1
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4answers
60 views

Co-domain & Image

I understand much so that the image is a subset of the co-domain of a function. It is also my understanding that the co-domain of a function is arbitrary, which would by extension mean whether or not ...
1
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1answer
28 views

rational integral with a quartic function in the denominator

Let's say I've got the integral: $\int [ 2 Q^4 - 5 Q^2 + 3 ]^{-1} dQ$ This integral evaluates to: $\int [ 2 Q^4 - 5 Q^2 + 3 ]^{-1} dQ = \tanh^{-1}\left( Q \right) + \sqrt{ \frac{2}{3} } ...
0
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2answers
43 views

Once I found the region, How can I found the Area

I have to find the area of the region bounded by $y = 4x^2$, the tangent line at $(3, 36)$ and the $x$-axis. I found the equation of the tangent line, which is: $y = 24x-36$ Both functions meet at ...
1
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2answers
34 views

How to differentiate between “shapes” when given parametric equations?

Having a hard time understanding how to figure out how a function looks like when it's in parametric form. Here are two examples, just wondering if someone could help me develop some sort of ...
0
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3answers
30 views

Name of this specific type of function

Is there a specific type of name for a function that requires its predecessor? Like so: $f(x+1) = 2f(x)$. I am using this for a report and have deduced a general function which doesn't require the ...
3
votes
2answers
35 views

What does this mean? (parabola)

The question is: If ($x,y$) represents a point on the graph of $y = 2x + 1$, which of the following could be a portion of the graph of the set of points ($x,y^2$)? The graphs are hard to put on ...
0
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0answers
7 views

Solving technique for bidimensional function

My math is a little bit rusty and this is my first question on this site, so sorry if I commit any mistake. I am trying to minimize this equation with WxMaxima: $f(x,y) = ...
0
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1answer
33 views

Linear function.

As an exercise, I'm trying to solve this question but I don't how to start. I would appreciate any form of help. A cell phone plan has a basic charge of £25 a month. The plan includes 400 free ...
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2answers
27 views

Set Theory and Function Questions.

I have two separate questions and have attempted them but I don't know if they're correct. I just guessed. I would like to know the method of working each one out please. ** Explain when the ...
0
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2answers
67 views

Prove $f=\cos(\vartheta)$ and $g=\sin(\vartheta)$ based on given assumptions

Let $f$ and $g$ be differentiable real functions on an interval $I$. Suppose that $f^2+g^2=1$ and that $f(0)=\cos\vartheta_0$ and $g(0)=\sin\vartheta_0$, where $\vartheta_0 \in \Bbb R$. If ...
2
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1answer
35 views

If card(A) > card(B) then there is no surjection from B onto A, w/o AC. [duplicate]

I am trying to prove that if cardinality of A is greater than cardinality of B, then there is no surjection from B onto A. Or any logically equivalent implication. Formally, card($A$) > ...
1
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1answer
50 views

Would $-3 ^{-x}$ be an exponential decay of growth?

Would $-3 ^{-x}$ be an exponential decay of growth? Any and all help appreciated.
6
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0answers
148 views

How to invert this expression involving $\tanh^{-1}$?

I've got the expression: $ x = \tanh^{-1}(p) - \sqrt{\frac{2}{3}} \tanh^{-1}\left( \sqrt{\frac{2}{3}} p\right) $ How can I invert this function so I have a function $p(x)$? I thought about using ...
3
votes
1answer
47 views

What is the domain of $\frac{1}{f(x)}$ in this case?

Suppose $f(x) = \frac{x-1}{x-2}$ , Then $\frac{1}{f(x)} = \frac{1}{\frac{x-1}{x-2}}$ So would the domain be all real numbers excluding 1 and 2 or would the domain include 2? Since ...
3
votes
2answers
72 views

Is the closed interval $[-\dfrac{\pi}{2}, \dfrac{\pi}{2}]$ homeomorphic to $\mathbb{R}$? [duplicate]

The open interval $(-\dfrac{\pi}{2}, \dfrac{\pi}{2})$ is well known to be homeomorphic to $\mathbb{R}$ through the homemorphism $f(x) = \tan(x)$ Is the closed interval also homeomorphic to the real ...
0
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1answer
28 views

Is the intermediate value theorem saying that if $f$ is continuous on some interval, then it is a surjective function?

Is the Intermediate Value Theorem basically saying that if a function is continuous on an interval, then the function is surjective? The formal definition states something to the effect of "any ...
3
votes
3answers
76 views

Prove that $ f(a+b) \ge f(a)+f(b)$.

Let $f$ be a function such that $f^{\prime \prime} (x) \ge 0$ , $f(0)=0$, for each $x \ge 0$. Prove that if $a,b \ge 0$ , then: $$ f(a+b) \ge f(a)+f(b)$$ It's really hard for me to get an ...
0
votes
4answers
69 views

$g\circ f$ bijective iff $f$ and $g$ bijective? [duplicate]

Is the following true: $g\circ f$ bijective iff $f$ and $g$ bijective? Or can the requirements be weakened for $g$ (i.e. $g$ only injective or surjective)? Or $f$?
1
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3answers
65 views

How to find when $f(x)>h(x)$

I got two functions $f(x) = 3x+x(x+1) + 610$ $h(x) = 817 + x$ How to calculate when $f(x) > h(x)$ In this case the answer is $x > 12$, but how do I calculate this? I been trying to compare ...
0
votes
0answers
16 views

Finding the radius of the resulting 3D solid

If I rotate the these functions about the y-axis, how can I find the radius of the resulting solid? $$x = 2\sqrt{2y}$$ $$x = 0$$ $$y = 5$$ I am trying to find the radius of the $3$ dimensional ...
0
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0answers
29 views

Find the minimum function

During my work I came to some problem, please take a look on the next table. ...
0
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0answers
19 views

Function with absolute value in denominator - limits

f(x)=(x-1)/(|2-x|-1) |2-x|= { |2-x|; x < +2} {-|2-x|; x >= +2} State domain, range and the equations of the asymptotes. D(f)= {x | x > 3 or x < 3} R(f)= {y | y > 1 or y <= -1} ...
3
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3answers
858 views

Function with infinitely many right inverses?

I was thinking about a function with infinitely many right inverses but I could not come up with anything. Does there exist a function with infinitely many right inverses?
0
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1answer
24 views

How do I determine the graph of functions involving radicals?

What is the explanation behind: the graph of $h(x)=\sqrt{4-x^2}$ is the upper half of the graph of $x^2+y^2=4$ the graph of $g(x)=-\sqrt{2-x}$ is the lower branch of the parabola $x=2-y^2$ I kind ...
0
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3answers
36 views

What behaviour will f(X) show?

Given $f(x)\cdot f'(x)<0$ for all x, then what will be necessarily true for f(x). $f$ is decreasing function $f$ is increasing function $\left|f\right|$ is a decreasing function $\left|f\right|$ ...
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1answer
41 views
1
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1answer
19 views

Estimates for functions with polynomial growth

Suppose function $g(x):\mathbb{R} \to \mathbb{R}$ satisfies the following polynomial growth condition: $$|g(x)^{(j)}| \leq K(1 + |x|^{p-j}), \quad j = 0,1,2,3$$ for some constants $K>0, p\geq 3$, ...
1
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5answers
115 views

If $0<x<1$ then prove that $x^a \leq x < 1$ for all $a\in \mathbb{R}, a\geq 1$.

If this is true then can somebody please help me to get the proof. Thanks! I am trying to see the proof of this for all $a\in \mathbb{R}, a\geq 1$.I saw that this is easy inequality to prove if $a\in ...
1
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1answer
16 views

Identification of Independent and Dependent Variables

I often have trouble identifying what is the dependent variable and what is the independent variable in written English. Consider this: Dependence of A on B. In this case it's clear A is dependent ...
3
votes
1answer
43 views

Name of property: $g(f(a), f(b)) = f(g(a, b))$

What's the name of the property of the functions $f$ and $g$ that lets one do this: $g(f(a),f(b))=f(g(a,b))$ For example, I'm looking for a certain class of functions that do this: $f(a) \oplus ...
2
votes
1answer
66 views

Is there an holomorphic function? If that function exists, is it unique?

I am solving a problem that asked me if exist an holomorphic function which satisfies only two condition. both equally: $$ f\left(\frac{1}{\alpha n} \right)=0\ \ \ \ and\ \ \ f\left( \frac{1}{\alpha ...
0
votes
1answer
20 views

How can I find the point of intersection between these functions?

Find the point of intersection between $f(x) = x^3$ and $g(x) = x^{1/3}$ Once I equal each equation to each other, I could factor out the $x$ but the exponent 1/3 is confusing me. Thank you!
3
votes
2answers
49 views

Finding Laurent's series of a function

I am trying express the function $$f(z)=\frac{z^3+2}{(z-1)(z-2)}$$ like a Laurent's series in each ring centering in $0$, but I do not now how could I express it, in first I said that ...
0
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1answer
53 views

Proove that the sequence $\left (1+\frac{1}{x} \right )^{x}$ increases. [duplicate]

I want to show that $\left (1+\frac{1}{x} \right )^{x}$ increases. I have to show that $\left (1+\frac{1}{x+1} \right )^{x+1} > \left (1+\frac{1}{x} \right )^{x}$ $\left ...
5
votes
2answers
33 views

How to make sense of $(1-e^{tD})f$?

I'm sophomore student in college. Recently, I'm thinking about series expansion of operators. When I supposed that f is an $C^\infty$-function and D is the differential operator d/dt. According to ...
1
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1answer
16 views

partials of a term resulting in same RHS [closed]

Does a term V exist $\ni$ $$ \frac{\partial\,V}{\partial\,x}= c_{1}\,x\,y $$ $$ \frac{\partial\,V}{\partial\,y}=c_{2}\,x\,y $$ where $c_{1}$ and $c_{2}$ are constants and $c_{1} \ne c_{2}$
0
votes
1answer
30 views

Average Velocity over a time interval

A Honda Civic travels in a straight line along a road. Its distance x from a stop sign is given as a function of time t by the equation $x(t)= αt^2− βt^3$, where$ α = 1.45 m/s^2$ and $β = 0.055 m/s^3$ ...
1
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0answers
9 views

Directrix and foci relationship on a hyperbola

on a parabola the distance from the vertex is equal to the distance from the directrix. Is this the case with hyperbolas? I have looked on multiple math websites and they don't state this but from ...