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

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2
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3answers
18 views

(Non)Existence of limits

When we say that a limit of a function does not exist in $\mathbb{R}$ (or some metric space) does it make sense to say that it might exist somewhere else? [I am trying to think along lines of ...
-1
votes
2answers
38 views

Finding the range of the function $\frac{3}{f(x)}$ [on hold]

Given that ${f(x)=2x^2+4x+3}$, find the range of the function $\frac{3}{f(x)}$.
8
votes
5answers
460 views

Is there a way to prove this exponential inequality?

I came across this proposition while trying to prove that a function was injective: if $a>b$ then $a^a>b^b$, where $a$ and $b$ are real numbers bigger than 1 . Intuitively it (somehow) makes ...
0
votes
1answer
24 views

Partial derivatives of $x$ and $y$ are equal at $(a, b)$ and $(b, a)$

If $f(x, y) = f(y, x)$ for all $(x, y) ∈ \mathbb R^2$. Show that $\frac{∂f}{∂x}(a, b) = \frac{∂f}{∂y}(b, a)$ For all $(a,b) ∈ \mathbb R^2$. So far I think that I need to put $f(x, y) = gf(x, y)$ ...
0
votes
0answers
30 views

How can show that the following function is non-negative?

I was working on a problem and reduced it to show the following inequality: ‎‎ $$\sum_{\substack{i,j=1\\i<j}}^{n}x_i^{\alpha}x_j^{\alpha}(\ln x_i-\ln x_j)^2+A_1\Big[\sum_{i=1}^{n}x_i^\alpha (\ln ...
5
votes
4answers
236 views

Finding coefficient of polynomial?

The coefficient of $x^{12}$ in $(x^3 + x^4 + x^5 + x^6 + …)^3$ is_______? Somewhere it explain as: The expression can be re-written as: $(x^3 (1+ x + x^2 + x^3 + …))^3=x^9(1+(x+x^2+x^3))^3$ ...
1
vote
1answer
33 views

Identifying the limit in the function

We see in the third part part that limit of 2$\theta$ is between -$\frac{\pi}{2}$ and -$\pi$ which become $-\frac{\pi}{2}<\pi + 2\theta<0$. How do we know that $2\theta$ became $\pi + ...
1
vote
1answer
27 views

The maximum possible size of $R$ is_____?

A function $f : N^+ → N^+$, defined on the set of positive integers $N^+$, satisfies the following properties: $f(n) = f(n/2)$ if $n$ is even $f(n) = f(n+5)$ if $n$ is odd Let $R = \{i|∃ j : f(j) = ...
3
votes
2answers
64 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 ...
2
votes
4answers
76 views

Suppose a function is expressed by: $f(x)=f(x+1) - f(x-1)$ and $f(16)=20 , f(20)=16$ What is $f(20162016)$?

Math quiz bee question Suppose a function is expressed by: $$f(x)=f(x+1) + f(x-1)$$ and $$f(16)=20 , f(20)=16.$$ What is $f(20162016)$? Attempt at solution: $f(17)+f(15)=20$ ...
7
votes
1answer
4k views

Difference between functional and function.

I have come across the term 'functional'. How is a 'functional' different from a 'function'? The exact term I came across was 'statistical functional.' In terms of the background, can you please ...
2
votes
0answers
77 views

Prove that the following function is convex?

I am trying to prove that the function $$g(\alpha)=\ln\Big(\sum_{i=1}^{n}t_i^\alpha+A(\alpha)\Big) ~~t_i, \alpha>0,$$ where $A(\alpha)=\int_{a}^{b}x^{\alpha+\beta-1}e^{-\lambda x^\beta}\,dx$,is ...
0
votes
1answer
27 views

Logarithmic function transformations

The standard log function form is $a \log[k(x-d)] + c$ Where $a$ vertically stretches or compresses $k$ horizontally stretches or compresses $d$ translates left or right $c$ translates up or ...
2
votes
0answers
26 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 ...
0
votes
1answer
20 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
33 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
0answers
40 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
2answers
73 views

Range of inverse harmonic mean of two integers

Today I was solving an exercise and one of the things I tried (which later turned out to be useless) involved considering the following: Is there a simple way to describe in terms of $n$ the range of ...
0
votes
0answers
22 views
0
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0answers
26 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
1answer
54 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
48 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
72 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
44 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 ...
3
votes
4answers
78 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
3answers
34 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 ...
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
1answer
23 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
64 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
32 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} ...
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 ...
0
votes
2answers
39 views

Finding a two-variable function that is distinct from another on every open disk, with specifics.

Consider the two-variable function $$f(x, y) = \sin(x) + \cos(x) + y^2.$$ Find a two-variable function $g(x, y)$ that is distinct from $f(x, y)$ on every open disk which contains the point $(1, 2)$ ...
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
vote
0answers
31 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 ...
1
vote
2answers
63 views

How to add/subtract complex rational expressions?

I'm studying for my Precalculus final and have noticed I still don't fully grasp performing basic operations on complex rational expressions, or finding if any values must be restricted from the ...
5
votes
4answers
84 views

Codomains and the definition of a function

A function $f$ is defined as a set of ordered pairs $(x, y)$ such that $(x,b), (x,c) \in f \Longrightarrow b=c$. Since $y$ is determined uniquely by $x$, it is customarily denoted $f(x)$. One ...
1
vote
1answer
21 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?
-1
votes
1answer
24 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 ...
0
votes
0answers
17 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 ...
0
votes
1answer
45 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, ...
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 ...
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
0
votes
2answers
42 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 ...
-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?
2
votes
3answers
26k views

If g(f(x)) is one-to-one (injective) show f(x) is also one-to-one (given that…) [duplicate]

Possible Duplicate: Injective and Surjective Functions If $g(f(x))$ is one-to-one (injective) show $f(x)$ is also one-to-one given that $f$ is a function from $A$ to $B$ and $g$ a function ...
1
vote
1answer
88 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
1answer
22 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 $\to$ ...
102
votes
8answers
4k views

Continuous function $f:\mathbb{R}\to\mathbb{R}$ such that $f(f(x)) = -x$?

I've been perusing the internet looking for interesting problems to solve. I found the following problem and have been going at it for the past 30 minutes with no success: Find a continuous ...