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

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0
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1answer
19 views

Find the domain of the given function.

I have the function $f(x)=\cos^{-1}(\frac{1}{2\cos(x)})$ and I have to find its domain. What I know is that the domain of $\cos^{-1}(x)$ is $[-1,1]$ so I think that $\frac{1}{2\cos(x)}$ should be at ...
0
votes
0answers
26 views

Proving an inequality involving integrals?

I am trying to prove that $$[\sum_{i=1}^{n}(\ln t_i)^2 t_i^\alpha+A^{\prime \prime}(\alpha)][\sum_{i=1}^{n}t_i^\alpha+A(\alpha)]\ge[\sum_{i=1}^{n}(\ln t_i) t_i^\alpha+A^{\prime}(\alpha)]^2$$ where ...
0
votes
1answer
35 views

is the followng function $f$ surjective?

$f$ is a function mapping $x$ axis to plane $V$ defined by if $P(x,0)$ then $f(P) = (x,x^2)$ ? I am not so sure for my following answer : to show that $f$ is surjective, for every $A(x,x^2)$ there ...
0
votes
0answers
22 views

In order to show or refute: Given a real function $f$ and $a, b \in R$ then $a\leq b \Rightarrow f(a)\leq f(b)$, what should I regard?

Is it enough to show that $f$ is increasing or decreasing in any interval $I$ that contains both numbers $a$ and $b$?
0
votes
0answers
24 views

Existence of a continuous surjection

Does there exist a continuous surjection from R (set of real numbers) to [0,2*pi]? (the set is closed). It seems apparent to me that this should be affirmative, but I haven't been able to provide a ...
2
votes
0answers
25 views

Constructing an iterative “signature” function

Please pardon my rather crude description of this problem, I am not very adept at mathematical notation and language, but I will do my best to describe it in a way as to be understandable. I have one ...
2
votes
1answer
51 views

f: R → R and $|f'(x)| ≤ |f(x)|$ [duplicate]

Let $f: R → R $ be a function such that $f'(x)$ is continuous and $|f'(x)| ≤ |f(x)|$ for all $x ∈ R$ , if $f(0)=0$ the maximum value of $f(5)$ is My Attempt: I proved that $f'(x)=0$ for $x ∈ [0,1]$ ...
1
vote
1answer
43 views

Prove that the following function has a unique maximum?

I was working on a problem and reduced it to showing $$f(\alpha)=n\ln \alpha-\ln \left(\sum_{i=1}^n t_i^\alpha+\int_a^b x^{\alpha+\beta-1} e^{-\lambda x^\beta} \, dx \right) + (\alpha-1)\sum_{i=1}^n ...
1
vote
0answers
71 views

Evaluating a Difficult limit!

I have to evaluate a very complicated limit, I've done this task already but I wanna make sure I did it right. The function I have in my hands is $$ F(\omega)= \tanh \Big[a\cdot ...
4
votes
2answers
43 views

How can one determine if a function should have parenthesis around their argument?

I have noticed that there are a select few functions that are acceptable if their argument is not in parenthesis. For example, here are a few functions I noted do not require an arguement: Trig or ...
0
votes
3answers
33 views

Inverse Trig and infinite values (arccos)

I understand that trig ratios can have infinite values for the same value of $x$ $ \cos(x) $ for example. Since $ \cos(x) $ shows the relationship between two sides of a triangle and that ratio can ...
3
votes
4answers
72 views

If $B\subset A$ and $f:A\to B$ is injective prove it's a bijection between $A$ and $B$

I want to show that if $B\subset A$ and $f:A\to B$ is an injective function then there's a bijection between $A$ and $B$. I believe my "proof" is wrong, I probably use too much "intuition" when I ...
0
votes
1answer
22 views

Problem with convolution, insecure

$$f(t)= t^2\cdot u(t),\quad g(t)=t^4\cdot u(t)$$ I know that I need to use convolution theorem to solve this problem, but I really don't know what to do with step functions. Do I need to include ...
1
vote
1answer
62 views

What type of discontinuity is found in this graph?

$$ f(x) = \begin{cases} \dfrac{1}{x} && \text{when $x > 0$}\\ 4 && \text{when $x < 0$} \end{cases} $$ What type of discontinuity is present when $f(0)$ ? ...
0
votes
2answers
30 views

which of the following are true out of the four statements? [on hold]

let $p(x)= x^n+\sum\limits_{i=0}^\mathbb{n-1}a_k x^k$ and $q(x)= x^n+\sum\limits_{i=0}^\mathbb{n-1}b_k x^k$ be two polynomials with real coefficients such that $n\geqslant 4$ is even and $a_{n-1} ...
3
votes
2answers
50 views

Construct $f: X\to Y$ such that $f(p)=p$

Let , $X=[-1,1]\times [-1,1]$ and $Y=\{0\}\times \left[-\frac{1}{2},\frac{1}{2}\right]$. Construct an example of a continuous map $f:X\to Y$ such that $f(p)=p$ for each $p\in Y$. I construct a ...
1
vote
1answer
13 views

showing a function is surjective for isomorphisms

Consider a problem like the following. Prove that $ \mathbb{R} $ is isomorphic to the ring S of all $ 2 \times 2 $ matrices of the form $ \begin{bmatrix}a & 0\\0 & a\end{bmatrix}$, with $a ...
1
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0answers
30 views

Finding convolution of two functions?

1. Continuous Functions $x_1(t)$ and $x_2(t)$ definitions' link How to evaluate $(x_1∗x_2)(t)$ at $t = −T, 0, +T$ in terms of $T$ 2. Discrete Functions $x_1[n]$ and $x_2[n]$ definitions' link ...
0
votes
1answer
29 views

Predicates about functions in 1st order logic

Given the usual definition of function as a subset of $ D \times C $. What is the correct way to write "All functions $ f $ from $ D $ to $ C $ have property $P(f)$". This is both a question about ...
1
vote
1answer
32 views

Finding the number of solutions of the equation $2x^5- 6x^3 + 2x = 4x^4 - 6x^2 +1$ in the interval $I = [-2, 2]$

I have to find the number of solutions for the following equation on the interval $I=[-2,2]$ $$2x^5- 6x^3 + 2x = 4x^4 - 6x^2 +1$$ Now I know I have to put them all on one side and then use the ...
0
votes
0answers
15 views

Detailed proof (submersion) : show that the differential is surjective

I'm currently studying manifolds and wanted to have a detailed insight on a part of some proof. This might be very easy, but I can't find the good words to express the correct idea. My definition of ...
-1
votes
1answer
23 views

Investigation of continuity of a piecewise function [on hold]

Investigate continuity of the following piecewisely defined function: $$f(x)= \begin{cases} x & \text{if } x \in \mathbb{Z} \\ 0 & \text{otherwise} \end{cases}$$ where $\mathbb{Z}$ is the ...
1
vote
1answer
20 views

Why does Ln x start higher on the graph than log x? They follow eacthother but Ln is always higher.

Theres no answers on the internet, I would like to know why ln x stays higher than log x. Does it have to do with Base 10 of log? and the natural log of e with ln x?
0
votes
1answer
44 views

Prove that $f(x)>0$ near $x=0$

Given that $f(x) \in C^{5}$ and $$ f(0)=f'(0)=f''(0)=f^{(3)}(0)=f^{(4)}(0)=0 $$ $$f^{(5)}(0)>0 $$ Prove that $$f(x)>0$$ near $x>0$ I know that this can be proved with Taylor's expansion, ...
0
votes
1answer
55 views

Does every quartic polynomial of the form $(x+a)(x+b)(x+c)(x+d)$ where $a \neq b \neq c \neq d \neq 0$ have a distinct local and absolute minimum?

To me it seems like yes because it's composed of 4 linear factors so it would have four roots. Also it's not negative fourth degree, so therefore has two convex lumps in the function. Perhaps ...
0
votes
3answers
60 views

Given two real functions, $f$ and $g$, if $|f(x)|<1$ then $|g(f(x))|<g(1)$? Why?

It seems trivial for a certain $x$ but can we say it for all x?
-1
votes
1answer
51 views

Discrete Mathematics question regarding functions. [on hold]

Let $S = \{s_1,s_2,...,s_n\}$. How many functions are there with domain $S$ and target Z2? Of those functions, how many are one-to-one? How many are onto?
1
vote
1answer
87 views

Let $f$ be injective and discontinuous at some point $c$. Can its inverse be continuous?

$f$ is injective at an interval $[a,b]$, but discontinuous at some point $c$ in the same interval. I need to prove that its inverse is continuous at that interval. Should I consider what is the ...
0
votes
2answers
45 views

Let f be continuous. By EVT there exists a c such that f(c)=supx f(x). Show that f is not injective.

I am given a continuous function f in an interval [a,b]. To show that f is not injective, should I consider the definition of the extreme value theorem? I am not sure how to show that it is not one ...
0
votes
2answers
41 views

$\int\limits_0^1 {\left( {1 - 2{x^2}} \right)f\left( x \right)dx}<0$, when $f$:convex and differentiable with $f(0)=0$

Let $f:[0,1]\to\mathbb{R}$ be a differentiable function that is convex and $f(0)=0$. Prove that: $\int\limits_0^1 {\left( {1 - 2{x^2}} \right)f\left( x \right)dx}<0$. I thought that since ...
0
votes
1answer
21 views

How to find $x$-intercept on TI-83/TI-84 calculator without having to set the bounds for each intercept?

Finding $y$-intercepts is very easy on a TI-83/TI-84 calculator. All you have to do is graph the function and use 2nd $\to$ ...
0
votes
0answers
17 views

Solving $f(x) \leq 10 f(kx) + 10kg(x)$ for $f, g$ nonnegative on $(0, 1]$

Suppose we are given two nonnegative functions $f$ and $g$ on $(0,1]$ that satisfy $f(x) \leq x^{-1/2}$ and $$f(x) \leq 10 f(kx) + 10kg(x)$$ for all $k$ sufficiently large. Is it possible to reduce ...
1
vote
0answers
33 views

Represetation of a smooth function in the neighborhood of its zero-set

Consider $k$ smooth functions $g_i(x)$, $x\in \mathbb{R}^n$, $k<n$. The set $G$ is defined as $G=\{x\in \mathbb{R}^n|g_i(x)=0, i=1,...,k\}$. We also assume that the Jacobi matrix $\frac{Dg}{Dx}$ is ...
-1
votes
1answer
42 views

CALCULUS: Sketching a function by given conditions [on hold]

Pls help. I'm currently on a struggle with this calculus problem. Thanks in advance.
0
votes
1answer
37 views

Meaning of square root in this situation

I wasn't someone who really paid attention in math class but grew very fond of it throughout the time I was out of school. Square Root = A number that produces a specified quantity when multiplied by ...
1
vote
1answer
17 views

Product of real-valued functions on $\mathbb{R}^n$

Let $f,g:\mathbb{R}^n\to \mathbb{R}$. What is $fg$? Is it function $\mathbb{R}^n\to \mathbb{R}$ or $\mathbb{R}^{2n}\to \mathbb{R}$? For example, taking $n=2$ let $f(x_1,x_2)=x_1+x_2+x_1x_2$ and ...
1
vote
1answer
12 views

Continuous functions on compact group and uniformity

If $G$ is a compact abelian group and $f\in C(G)$. Then $\forall \epsilon >0$,there exists an open neighbourhood $U$ of $0\in G$, such that $\forall g\in G , \forall u_1,u_2\in U$, we have ...
0
votes
0answers
37 views

$f(x) = 2x \mod 1$ not equal to zero for all $x$?

If any number $\mod 1$ is zero, then how can $f(x) = 2x \mod 1$ be a Baker's map? For any $x\in \mathbb{R}$, shouldn't $f(x)=0$?
1
vote
1answer
33 views

Prove that $ϕ ◦ ϕ = ϕ^2 = Id_{\Bbb C}$ (the identity map on C) if and only if $e^{iθ} \bar c + c = 0$.

Consider the isometry $ϕ : \Bbb C → \Bbb C$ with equation $ϕ(z) = e^ {iθ} \bar z + c$ where $θ ∈ \Bbb R$ and $c ∈ \Bbb C$. Prove that $ϕ ◦ ϕ = ϕ^2 = Id_{\Bbb C}$ (the identity map on C) if and only if ...
0
votes
0answers
40 views

Prove closed disc $D^n$ is homeomorphic to the cone $CS^{n-1}$

I need to find a continuous surjective map from $D^n$ to $CS^{n-1}$. For 2 dimensions, we can use $$f: S^1 \times I /S^{1} \times \{1\} \rightarrow D^2$$ with $f(\theta,t) = (1-t)e^{i \theta}$ ...
1
vote
2answers
27 views

Constancy of an integral function

Fix some $\ell\in\mathbb{R}^+$. Say that $f:\mathbb{R}^2\to\mathbb{R}_{\geq0}$ and $\mu:\mathbb{R}\to\mathbb{R}^+$ are functions satisfying the following: $f$ and $\mu$ are continuous. $f$ is ...
3
votes
3answers
48 views

Find the area of the region described by $|5x|+|6y| \le 30 $

Find the area of the region described by $|5x|+|6y| \le 30 $ (where $|z|$ denotes the absolute value of $z$). My effort Imagining a number line and interpreting the problem as the request to ...
0
votes
1answer
33 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: ...
2
votes
2answers
48 views

Find $f$, when $f(1)=f(2)=1$ and $f'(x)\leq (x^2-x-1) e^x, \forall x\in [1,2]$

Let $f:[1,2]\to\mathbb{R}$ be a differentiable function such that $f(1)=f(2)=1$ and $f'(x)\leq (x^2-x-1) e^x, \forall x\in [1,2]$. Find $f(x)$. I am pretty sure that from things I have tried ...
1
vote
0answers
12 views

Determining function of Graph

Looking through some notes on numerical computation I came across the following graph: I know this is a longshot, but I'm not incredibly mathsy and would like to know what kind of a function this ...
0
votes
1answer
64 views

A weaker than “zero derivative” condition implies that a function is constant?

Let $f:(a,b)\to R$ be a continuous function such that $\limsup\limits_{n\to \infty}\frac{f(x_n)-f(x_0)}{|x_n-x_0|}\leq 0$ for every $x_0\in(a,b)$ and sequence $x_n$ converging to $x_0$ such that ...
2
votes
3answers
44 views

Taylor expansion of $\cos{x}$

I found a pdf file on the internet which gives you known expansions of Taylor's. There is something I cant understand : Why is the remainder of $\cos x$ is written like this? $$\frac{\cos ...
3
votes
3answers
33 views

Determine whether $e^x(1-e^x)\leqslant (1/4)$ for $x\lt 0$ is true or false?

The statement is $e^x(1-e^x)\leqslant (1/4)$ for $x\lt 0$ I think the above statement is true by calculating approximate value at some points. But how to prove it properly?
0
votes
1answer
19 views

Discrete Math Compositions

I am having trouble with these compositions. $$T = \{(a,a), (a,b), (b,c), (b,d), (c,d), (d,a), (d,b)\}$$ $$U = \{(a,a), (a,d), (b,c), (b,d), (c,a), (d,d)\}$$ I need to find $T \circ T$, $U \circ T$, ...
-1
votes
1answer
52 views

Finding the domain of $\frac{1}{x}|x^2 - 1|$ [on hold]

What is the domain of this function $F(x)=\frac{1}{x}|x^2 - 1|$ Can someone please tell me how to find it ?