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

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0
votes
2answers
23 views

Proof about composed functions

Let $f\colon X \to Y$ and $g\colon Y \to X$ be functions. Assume $g \circ f$ is bijective. Prove $f$ is injective and $g$ is surjective. Approach: if $g \circ f$ is bijective then $g \circ ...
4
votes
2answers
176 views

Find the domain

I have been a bit confused about finding the domain of these functions. 1) $\dfrac{12}{2x+3}$ 2) $\dfrac{4x-3}{x^2-81}$ 3) $\dfrac{x^2 -3x -18}{x-6}$ So I solved for $x$ and then those were the ...
-1
votes
1answer
31 views

Prove that g:R$\to$ (-1,1] defined by g(x)= cos x is surjective or not?

Surjective: a function $f$ from $A$ to $B$ is called onto or surjective, if and only if every element $b \in B$ there is an element $a \in A$ with $f(a)=b$. Really appreciate if someone explain ...
0
votes
0answers
12 views

Show that $C/B \approx T/A$

Let $T=S^1 \times S^1$ be the torus with meridian $A=S^1 \times \{1\} \subset T$ Let $C=S^1 \times [-1, 1]$ be the cylinder with base circles $B=S^1 \times \{-1, 1\}$. Show that $C/B \approx ...
0
votes
0answers
18 views

If f:[0,∞)→R is a **continuous and bounded** function -> so f has a maximum, minimum or both

I need to find an example that contradicts this next sentence: If f:[0,∞)→R is a continuous and bounded function -> so f has a maximum, minimum or both Any tips about how to solve this question? ...
0
votes
1answer
23 views

What is the full width of a peak of the function $F(X)=\frac{1+\cos((2N+1)πX)}{1+\cos(πX)}$

With $$1 + \cos \theta = 2 \cos^2 \frac{\theta}{2},$$ the function becomes $$f_n(x) = \left( \frac{\cos \frac{(2n+1)\pi x}{2}}{\cos \frac{\pi x}{2}} \right)^2.$$ It peaks at odd X integer values. ...
0
votes
2answers
36 views

Relation between $\int_{a}^{b} f(x) dx$ and $\int_{a}^{b} (1-f(x)) dx$

Say you're expected to work out $\int_{0}^{\pi/3} \sin^2(x) dx$ solely from the result $\int_{0}^{\pi/3} \cos^2(x) dx$. It can be transformed into $\int_{0}^{\pi/3} (1-\cos^2(x)) dx$, but then what?
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 ...
0
votes
1answer
38 views

$n$th derivative of function $\frac{1}{(1-2x)^2}$

I am trying to find the $n$th derivative of the function $\frac{1}{(1-2x)^2}$. The first three are simple but I can't see a schema right now. \begin{align*} y^{\prime} & = \frac{4}{(1-2x)^3}\\ ...
2
votes
2answers
35 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 ...
1
vote
0answers
51 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 ...
4
votes
3answers
160 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 ...
-3
votes
2answers
50 views

How do I develop a formula to find the height of a male using his femur? [on hold]

I'm a grade eleven math student and I need a formula to find the height of a 15/16 year old male subject using only the length of his femur. I'm very confused on how to do this, please help, it's for ...
4
votes
4answers
548 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 ...
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
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 $f$? ...
1
vote
0answers
45 views

Cumulative distribution function of two independent and uniform on $[0,1]$ random variables is a surjective map for $t\in [0,1]^2$?

I am trying to argue that the cumulative distribution function of two independent and uniform on $[0,1]$ random variables is a surjective map for $t\in [0,1]^2$. Below the argument I have developed. ...
3
votes
2answers
66 views
+50

Jensen-like averaging inequality on integers

Let $\mathbb{Z}^*=\mathbb{Z}^+\cup\{0\}$. Let $f:\mathbb{Z}^*\rightarrow\mathbb{R}$ be a nondecreasing function such that $f(a+b)\leq f(a)+f(b)$ for all $a,b\in\mathbb{Z}^*$. Is it true that for all ...
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
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
1answer
59 views

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

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
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 ...
-2
votes
1answer
67 views

The range of a twisted composition of sines [on hold]

Let $f(x) = \sin \left( \frac \pi 6 \sin \left( \frac \pi 2\sin x \right) \right)$ and $g(x) = \frac \pi 2 \sin x$ for all $x \in \Bbb R$. Which of the following are true? The range of $f$ is ...
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 ...
1
vote
1answer
45 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 ...
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 ...
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 ...
4
votes
4answers
77 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
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
1answer
19 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). ...
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
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 ...
2
votes
2answers
33 views

Is a function of a function the same as a two-argument function?

Say I have a function $g$ described by $$(x\stackrel f\mapsto y)\stackrel g\mapsto z$$ That is, $g$ takes a function, $f$, and maps it to another object, maybe just a real number or vector or ...
0
votes
0answers
118 views

What is the range of this function?

What is the range of $h$? $f(x)=4x+1$ $g(x)=(x-1)/3$ Let $h=\{f^n(g^m(1)):n,m\in\mathbb{N}\geq0\}$ What is the range of $h$? Show that $(2\mathbb{N}-1)\subset H$. ... okay I've done a bit more: ...
0
votes
1answer
15 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 ...
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
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
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
22 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 $y = f(x)$ ...
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?
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
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
102 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
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 ...
-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 ...