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

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2
votes
1answer
43 views

Under what conditions on $f$, is $f(az)=g(a)f(z)$?

Formal Statement Given nonzero constant $a \in \mathbb{C}$, $|a|>0$ and $f:\mathbb{C} \to \mathbb{C}$, under what conditions on $f$ does the following hold? \begin{equation} f\left(a ...
2
votes
2answers
36 views

Limit of indeteminate form $1^{(∞)}$

If we consider the function $f(x)=[(ax+1)/(bx+2)]^{x}$ where $a$,$b$ >$0$ and a I tried as follows]1 But at end i got stuck .
0
votes
0answers
38 views

Homeomorphism between $\mathbb{R^2}$ and $S^2-N$, the sphere without its north pole

How would one approach the following problem? Write down a homeomorphism and its inverse from $\mathbb{R^2}$ to the sphere $S^2-N$ without its north pole So I need a function $f(x,y) : ...
-1
votes
0answers
7 views

How can I find the necessary speed and speed of rotation for a problem from a parametric equation?

I have been given the following questions for a project that I am currently working on: Questions 1 to 8 I have completed questions 1 through 6 but have no idea how to do questions 6 or 7 after ...
4
votes
3answers
165 views

Why is $\log(1+e^x) - \frac{x}{2}$ even?

I'm dealing with Fourier series and I'm trying to figure out $\log(1+e^x) - \frac{x}{2}$ is even??? I've tried the $f(-x) = f(x)$ method but it doesn't give me the equality. But I've plotted it, and ...
1
vote
0answers
52 views

About a geometric algorithm to compute $\sin$ based on the unit circle

In an old post I have found a user which claims to have a geometric algorithm to compute trigonometric  functions for an angle between $0^\circ$ and $90^\circ$ based on the unit circle. Here's the ...
2
votes
6answers
530 views

Separation of variables

If I have a function $f(x,y)$, is it always true that I can write it as a product of single-variable functions, $u(x)v(y)$? Thanks.
4
votes
4answers
550 views

How are domain and co-domain of a function useful?

I'm at university and I learned linear algebra, set theory, logic, and other kind of mathematics that use functions a lot. Now, I know that function is very important and useful in mathematics but I ...
2
votes
2answers
59 views

How to “rotate” a function? Or, how to write a function which has a known, rotational symmetry with respect to another function?

EDIT 2: I've posted my "real" question here: http://mathematica.stackexchange.com/questions/115766/finding-closed-form-eigenvalues-of-a-particular-matrix I have posed my question formally in LaTeX ...
1
vote
1answer
22 views

$L^2$ - base functions

say we have a function, $f$, in $L^2$ and have different base functions for $L^2$, then is there a reason to believe that one base is "better" compared to the other when writing expansions for $f$? ...
0
votes
1answer
136 views

Prove an additive function has property f(x)=x

So I am given a function $f: \mathbb{R} \rightarrow \mathbb{R}$ such that $f(x+y)=f(x)+f(y)$ for all real $x$ and $y$, is continuous at $x=0$, and $f(1)=1$. I need to show that $f(x)=x$ for all real ...
0
votes
1answer
67 views

Create an equation for a description of a rational function

A graph has a $y$-intercept at $-5$, no $x$-intercepts, and discontinuous points at $(-1,-5)$ and $(3, -5)$. I want to form an equation for this graph, but I don't know how the $y$-intercept relates ...
3
votes
1answer
51 views

What might this function be?

The problem: I'm looking for a particular function $f(x, y)$—this isn't "homework" in the sense that I have no idea if such a function exists. It has a continuous domain $-1 \lt x \lt 1$ and $-1 \lt ...
1
vote
1answer
47 views

Express $y = KC^x$ as a linear function

Consider an exponential relationship of the form $y = KC^x$ where $K$ and $C$ are constants. Express the exponential function $y = KC^x$ as a linear function and describe how you would obtain the ...
0
votes
1answer
40 views

Continuous function, finding its value [duplicate]

If a function $f: \mathbb{R}\to\mathbb{R}$ is continuous and $f(x+y) = f(x) + f(y)$ for all $x,y\in\mathbb{R}$, then what is this function $f(x)$?
0
votes
1answer
59 views

Finding value of functions $f(x) g(x)$ [closed]

If $f(x)=2x^3+4x^2+3x+2$ and $g(x)=2x^3+x^2+4$, where $f(x), g(x) \in \mathbb{Z}_5[x]$ then $f(x) g(x)$ is equal to ?
0
votes
0answers
17 views

Integral of implicit function - geometric meaning [on hold]

What is the geometric meaning of implicit function $$f(x,y) = 0$$ integral? Is it the same as for explicit function, eg. area, volume etc.. and we are computing it for the $0$ result, or is there ...
0
votes
0answers
15 views

Monotony and convexity of $U(t) = w(t) - t. w'(t)$

Let $\mathcal{D} = (\mathbb{R}^{+*})^2$. $c \in ]0,1[$. Moreover we have $\theta < 1$ and $\theta \ne 0$. We consider the following function (which is called CES or constant elasticity of ...
0
votes
1answer
22 views

how to check one-one and onto correspondence between two sets in Real

Let $A=\{x∣0<x<1\}$ and $B=\{x∣1<x<2\}$. Which of the following statements is true? There is a one to one, onto function from A to B. There is no one to one, onto function from A to B ...
0
votes
0answers
28 views

A question on composite functions

Let $f:[a,b]\to\mathbb{R}$ and $g:[a,b]\to\mathbb{R}$ be two functions such that $f′(x)=g′(x)$ for all $x \in [a,b]$, where $f′$ and $g′$ denote the first derivative of $f$ and $g$, respectively. ...
0
votes
0answers
28 views

How to identify the closest values multiple of 96?

I've a list of 8820 values spreaded in the interval [0, 1[. Thus, 1/8820 * t, with t ...
4
votes
4answers
78 views

Determine whether the function $f(x) = \cos x$ from $\mathbb{R}$ to $\mathbb{R}$ is surjective?

I am working on this question first I want to understand the question itself, what was the question asking me? For me, I think $\mathbb{R}$ to $\mathbb{R}$ are real numbers and if $\mathbb{R}$ to ...
1
vote
1answer
23 views

Finding the points of intersection

Let f(x) be a real valued function defined for all real numbers s such that $|f(x)-f(y)| \leq (1/2)|x-y|$ for all $x,y$. Then what is the number of points of intersection of the graph of $y = f(x)$ ...
1
vote
2answers
35 views

If $x$ is an integer such that $\operatorname{gcf}(x, 24) = 8$ and $\operatorname{lcm}(x, 24) = 312$, find $x$.

I'm confused with this problem. Any kind of help is appreciated. What are $\operatorname{gcf}(x,24)$ and $\operatorname{lcm}(x,24)$? Are they just another way to say $f_1(x)$ and $f_2(x)?$ What is the ...
1
vote
2answers
26 views

create function from graph/from limits

I am a Calculus I student and we are into our second week and finishing up limits. I know how to create a graph from limits, and I know that for example a parabola would match with a quadratic ...
23
votes
3answers
2k views

Given $f(f(x))$ can we find $f(x)$?

Given $f(f(x))=x+2$ does it necessarily follow that $f(x)=x+1$? This question comes from a precalculus algebra student.
1
vote
2answers
69 views

Is this notation on the restriction of a function in group theory common?

If $f: X \rightarrow Y$ is a function between sets $X$ and $Y$, then a common notation to use when we want to restrict $f$ to a certain domain $X' \subset X$ is $f|_{X'}: X' \rightarrow Y$. I'm doing ...
0
votes
1answer
28 views

Workbook recommendation in preparation for Electrical Engineering

I'm currently preparing myself for starting my graduate degree in Electrical Engineering. The mathematics courses given are outlined as follows: Mathematics 1 Real functions Continuity, limits, ...
-1
votes
2answers
30 views

How to find intersections of sine and cosine functions with $X$ axis

I've been struggling with this question for a few days, because I've been able to find the said intersections, but based on suppositions, rather than on mathematical process. For example, if I have ...
1
vote
1answer
20 views

Closed Graph Theorem on Finite Dimensional Banach Spaces

My professor wants us to prove that every linear mapping from a finite dimensional Banach space is continuous. BUT, he wants us to do so using the Closed Graph Theorem (in functional analysis). ...
1
vote
0answers
36 views

Proving analytic function $f = 0$ under certain assumtions

I was given the following exercise: Let $f(z)$ be analytic in an open and connected set $U$ containing the point $z=0$ and assume $|f(1/n)| < \frac{1}{2^n}$ for $n \in \mathbb{N}_{> 0}$. Prove ...
0
votes
1answer
16 views

Number of points which satisfy the tangency condition

The number of points in the rectangle {(x,y) $-10 \leq x \leq 10$ and $ -3\leq y \leq 3$ which lie on the curve $y^2 = x+sinx$and at the which the tangent to the curve is parallel to the x-axis, is ...
0
votes
1answer
19 views

Identifying the periodic function

Which of the functions below is not periodic ? a.) $e^{sinx}$ b.) $(10+sinx+cosx)^{-1}$ c.)$log(cosx)$ d.)$sin(e^x)$ My question - Although I could intuitively find out the answer to the ...
1
vote
0answers
26 views

How do we know that a function can be written as a power series?

Most proofs of a Taylor series or a Maclaurin series assume that the function can be written as a power series. If a function can be written as a power series then: $$f(x)=\sum_{n=0}^\infty ...
0
votes
1answer
22 views

Upper Bound of a Function defined on a Closed Interval

In my Textbook, I am given the follow function which is defined on the closed interval $[a,b] $ $$(1/21)\cdot(x*7-3x*4+x+4)\le 6/21$$ $$(1/21)\cdot |7x*6-12x*3+1| \le 20/21 $$ These functions are ...
-1
votes
0answers
40 views

Function Question, Tried my level best no answer! [closed]

Look at this: $$f(x) = \frac {2x(\sin x +\tan x)}{2\lfloor(x+2\pi)/\pi\rfloor-3}$$ where $\lfloor\, .\rfloor$ denotes greatest integer function. Then how is function odd?
2
votes
2answers
28 views

How can I prove the concavity of $f(p_1,p_2,\ldots,p_n) = \sum_{i = 1}^n p_i(1-p_i)$?

Assume $p_n$ is the probability of being in class $n$ which mean that $f(0) = 0$ , $f(1) =0$ , and $p_1+p_2 = 1$ I need to come up with a concave function that show the relation between $p_1$ and ...
0
votes
1answer
420 views

excel- function with multiple variables

So I'm not good with excel (computers in general) and can do some things but this one is out of my league. This is the problem: The cost of a used car is highly correlated with the following ...
0
votes
1answer
54 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 ...
1
vote
0answers
57 views

pre-image of intersection

This is a basic question and I know this is typically not how this is proven but I was wondering if the following is a valid proof of showing that given two disjoint sets, say V and W in the ...
1
vote
1answer
36 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 ...
0
votes
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 ...
-1
votes
0answers
48 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
2answers
102 views

Is torus w. disc removed homotopic to Klein bottle w. disc removed?

I know that homeomorhic spaces are homotopic, but am not sure if this applies, since I think they are not homeomorphic due to orientability. I know $f$ and $g$ are homotopic if they represent: ...
1
vote
1answer
41 views

Terminology for functions with $F(a,a,\dots,a) = a$

Is there a commonly used terminology for functions $F : \mathbb{R}^n \rightarrow \mathbb{R}$ such that if $x \in \mathbb{R}^n$ and $x_i = a$ for all $i\in \{1,\dots, n\}$, $F(x) = a$ ?
2
votes
1answer
54 views

How do we know which terms are of higher order?

From Asymptotic analysis and perturbation theory by Paulsen: Find the behavior of the function defined implicitly by $$x^2+xy-y^3=0$$ as $x\to\infty$. [...] At this point, we have shown ...
0
votes
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 ...
4
votes
1answer
41 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 ...
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 ...
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 ...