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

learn more… | top users | synonyms (1)

1
vote
4answers
127 views

Find domain of logarithmic function

I'm a little confused about this question , I know that fraction denominator needs to be > 0 , so if denominator is quadractic equation i know how to solve , But since denominator is > 0 , I dont know ...
2
votes
2answers
36 views

Find out if $\log(3x + \sqrt{9x^2 + 1})$ is even or odd

I'm trying to find if this function is odd or even : $f(x) = \log(3x + \sqrt{9x^2 + 1})$ I know that it's an odd one because if I try $f(x) + f(-x) = 0$ it shows that it's odd. But I want to know ...
1
vote
1answer
48 views

Which of the following cannot be the value of $g(x)$

Let A = $\begin{bmatrix}1 & \tan x\\-\tan x & 1\end{bmatrix}$ then let us define a function $f(x)=\begin{vmatrix}A^{T}A^{-1}\end{vmatrix}$ then which of the following cannot be the value of ...
0
votes
0answers
23 views

Proof that pull-back of closed set by continuous functional is weakly closed needs continuity?

Question: In Banach space $X$, if $\phi \in X^*$, then pullback of a closed set is weakly closed? I wrote the following proof: Let $X$ be Banach space. Let $I$ be a closed interval of ...
1
vote
2answers
20 views

How to find the equation of $f(x^3)$ given the tangent in $f(x)$? [closed]

The exercise says: The tangent of $f(x)$ in $x=1$ is $y=2x-1$. Find the tangent of $y=f(x^3)$ in $x=1$.
2
votes
1answer
31 views

Proving the existence of a bijection between $U_{mn}$ and $U_m \times U_n$ where $(m,n)=1$ , there by proving Euler $\phi$ is multiplicative

Without proving before hand that Euler's phi $(\phi)$ function is multiplicative , can we prove that there is a bijection between $U_{mn}$ and $U_m \times U_n$ , for any pair of relatively prime ...
0
votes
3answers
60 views

How to write a function to express not divisible by 3?

This problem is from Discrete Mathematics and its applications And the example problem that the author gave For 4a, I immediately recognized this set is countable and infinite because you could ...
-1
votes
2answers
38 views

What function to use to show one to one correspondence?

This problem is from Discrete Mathematics and its Applications Here's an example problem that the author gave I'am working on problem 2e. I first recognized the set as countably infinite. If you ...
-1
votes
1answer
30 views

Clarification on intuition behind one to one correspondence?

My book - Discrete Mathematics and its Applications This is my book's definition on if an infinite set is countable And the example it gave The "infinite set is countable if and only if it is ...
1
vote
1answer
30 views

How to display one to one correspondence?

This is a problem from Discrete Mathematics and its Applications Here is the book's definition of countable/not countable For 2a, I came up with the fact that the set is countably infinite. What ...
0
votes
0answers
22 views

linear algebra, transformations [duplicate]

Let $V$ an $n$-dimensional vector space and $T$ a linear operator on $V$. Suppose that there is some positive integer $k$ such that $T^{k}=0$. Prove that. $T^{n}=0$
0
votes
1answer
51 views

Is this piecewise function Riemann Integrable?

Is $f(x)$ where, $ f(x) = \left\{ \begin{array}{lr} \frac{1}{x^2} & : x <0\\ x & : x \geq0 \end{array} \right. $, Riemann Integrable over $[-1,1]$? ...
0
votes
2answers
43 views

Find a function value given 2 points

Given $f(x)$, which is differentiable at every point such that: $f'(x) \ge -5$ for every $ x \in R$ $f(2) = -13$, $f(9) = -48$ Prove that:$ f(3) = -18$ Now it's quite obvious that $ f(3) = -18$ ...
0
votes
3answers
40 views

Prove inequality of 2 functions for every $ x \gt 1 $

Prove that for every $ x \gt 1$ exists: $$ 2x^3 + x^2-􀀀2x < x^4+5x^2-5 $$ I got this on my calculus course at college, can I get some suggestions? I'm really breaking my head on this, and ...
0
votes
1answer
33 views

Are there any other functions that are always discontinuous?

Looking at a graph (or finding some solutions with a calculator) of the function $f(x)=n^x$, where $n<0$, shows that it has one interesting property, so far as I can see: It is discontinuous ...
0
votes
4answers
52 views

Show that function is bounded

Show that the function is bounded function I think it is not (from the graph), but how to prove this? the functions is $$\frac{1}{x^2 \sin x \ln x}$$
5
votes
4answers
189 views

How can we say two algebraic expressions are “equal” if one is undefined at certain points and the other isn't?

I'm trying to understand why it is that we can say $\frac{x^2-1}{x-1} = \frac{(x-1)(x+1)}{(x-1)} = x+1$ but then have it also be the case that the two functions $f(x) = \frac{x^2-1}{x-1}$ and ...
1
vote
2answers
102 views

What is cos²(x)?

This looks odd to me. I need a definition. Is it just the square of $\cos(x) $ ? Like $\ \cos^2(x) = \cos(x) \cdot \cos(x) $ ? Then why don't you write it like that: $\cos(x)^2 $ ?
0
votes
1answer
16 views

Bounded function with parameter

Let f be defined $f:(a,b) \rightarrow \mathbb{R}$, $f$ is continous and $a,b \in \mathbb{R}$. If $a=-\infty$ or $b=\infty$, is the function bounded? I do not know how to figure this out, detailes and ...
0
votes
2answers
19 views

Finding time left (With knowing duration and percentage) [closed]

Suppose I am given an unknown span of time, but I know that I have used up 69.56% of that time in 6:58:10 (HH:MM:SS). How can I calculate the total time of the span and the amount of time I have ...
0
votes
0answers
18 views

Uniform continuity of the function [closed]

Investigate the uniform coninuity of the function: $f:(0,1)\rightarrow \mathbb{R}$ $$f(x)=\frac{1-\cot \pi x}{x^2 \ln x}$$ I do not know how to start this question.
0
votes
2answers
18 views

Existence of an analytic function under some given conditions

Which of the followings is(/are) correct? There exists an entire function $f:\mathbb C \to \mathbb C$ which takes only real values & is such that $f(0)=0$ & $f(1)=1$. There exists an ...
1
vote
0answers
17 views

Given a $f : \mathbb{N} \rightarrow \mathbb{R}$ find $D \mathbb{R} \mapsto \mathbb{R}$ so that $f(x)=D(x+1)-D(x)$

Suppose you have a function $f(x)$, $f:\mathbb{N} \rightarrow \mathbb{R}$ now you want to find a function $D_f(x)$, $D_f: \mathbb{R} \rightarrow \mathbb{R}$ so that $f(x) = D_f(x+1)-D_f(x), \forall x ...
0
votes
0answers
19 views

continuity of the function with parameter

$f:(a,b)\rightarrow \mathbb{R}$ is continous $a,b\in \mathbb{R}$ Investigate the existance of a limit $\lim_{x\rightarrow a} f(x)$. Do not understand this utterly, please help.
0
votes
1answer
18 views

Unique expression as disjoint union of indecomposable subsets

Let $f:A \to A$ be a function, we say that $B \subseteq A$ is $f$-invariant iff $f(B) \subseteq B $. We say that an invariant subset is indecomposable iff it cannot be expressed as a union of ...
1
vote
1answer
28 views

Invertible “Sigmoid + x” function

I need an invertible function that represents a smooth transition between two straight, parallel line segments, like this: Depicted is $f(x) = -0.3/(1+e^{-10*(x-p)})+0.3/2+x$ (where $p$ is the ...
0
votes
1answer
12 views

$\{h\in A^B|h \text{ is invertible}\}$ is equiumerous to $\{k\in B^A|k \text{ is invertible}\}$ and $\aleph_0$ right invertibles for a function

1.Let $A,B$ be sets, prove: $\{h\in A^B|h \text{ is invertible}\}$ is equinumerous to $\{k\in B^A|k \text{ is invertible}\}$ 2.Let $A,B$ be sets and a function $f\in A^B$ give an example right ...
1
vote
4answers
43 views

Would this be an acceptable answer for the inverse of floor function

This problem is from Discrete Mathematics and its Applications And the book's definition on inverse Would an acceptable answer to 43b just be the set itself again? What I like to think of the ...
0
votes
2answers
37 views

Graph of $y=\text{constant}*x$

Graph of $y=x$ , $y=\frac{1}{2}x$ and $y=\frac{1}{4}x$ is All are straight line. But with different slope. Why?
5
votes
2answers
325 views

Is this expression even a function?

This is from Discrete Mathematics and its Applications My question is on 22c. From the book, I inferred from the question that all of the listed mathematical expressions are functions. Is the ...
0
votes
0answers
65 views

Linear algebra, linear functionals [closed]

$W^{0}=\left\{ f\in L\left( V,\Bbbk\right) ;f\left( v\right) =0,\forall v\in W\right\} .$ Yes is the Kunze. I'm also having trouble identifying. Do you have any more suggestions? Made the item ...
2
votes
1answer
83 views

Why is this not a function?

This problem is from Discrete Mathematics and its Applications This is the definition that the book gave of function Here is my work so far It's pretty clear to me that 1b and 1c are not ...
0
votes
0answers
9 views

Phase difference of two signal of different frequency

Currently, I have two signals, the main components of both signals are 60Hz, but both also have weaker response at 180Hz + small amount of noise. As shown in the photo below, I want to find the phase ...
2
votes
0answers
34 views

Is it true that Newton's Method converges quadratically always unless there's a multiple root?

I'm taking a look at some functions in Matlab and trying to decide how to find how fast roots converge during newton's method. I have a polynomial -- is it safe to say that if the function at the ...
0
votes
1answer
38 views

Applications for holomorphic functions?

Could anyone give me an insight into practical applications of holomorphic functions (I am using the term in the way in which it is related to Riemann's work)?
0
votes
1answer
32 views

Uniform continuity of two functions

Investigate uniform continuity of the following functions: $$a) \ f(x)=\frac{1}{x} \\ b) \ f(x)=\cos \frac{1}{x}$$ How to deal with such questions, i have little knowledge about that topic thus i ...
1
vote
1answer
17 views

Tell if a series converges uniformly

Let $f(x) = \sum_{n=1}^\infty \frac{-2x}{(x^2+n^2)^2}$. Check if $f_n(x)$ converges to a continuous function. So I've seen a solution that uses the fact that if $f(x)$ converges uniformly and ...
1
vote
1answer
18 views

Showing a function to be a norm

I want to prove or disprove that $\parallel (x,y)\parallel=\sqrt{\frac{x^2}{9}+\frac{y^2}{4}}$ is a norm on $\mathbb{R^2}$. Since $\{(x,y):\parallel(x,y)\parallel\leq1\}$ is a convex set, ...
0
votes
1answer
24 views

$f(x)$ be differentiable and have a local minimum in $x_0$, show with definition $f'_+(x_0)\ge0, f'_-(x_0)\le0, f'(x_0)=0$

Let $f(x)$ be differentiable in $x_0$, $x_0$ is a local minimum, prove with the definition that $f'_+(x_0)\ge0, f'_-(x_0)\le0, f'(x_0)=0$. I get that $f$ is decreasing from the left and ...
1
vote
0answers
20 views

The maximum notation as regards the absolute value?

We know that $\max(\textbf{A})$ gives the maximum element of the array $\textbf{A}$. What is the notation, or a short formula, if we seek the element having the largest absolute value? e.g., ...
1
vote
3answers
36 views

Size of function spaces

For example, how big is the space $ C^k(\mathbb{R},\mathbb{R}) $ ? How much is, say, $ C^0 $ larger than $ C^1$ ? How can one figure out ?
1
vote
2answers
40 views

What is the behavior of these functions linear or logarithmic or neither?

Please consider the following functions $F$ and $G$. \begin{align*} F(K) = \log_2 \left(\frac{( 2\sqrt{K-1}+K-2)^2}{(\sqrt{K-1}-1)^2}\right)+(K-1) \log_2(2) \end{align*} and the function ...
0
votes
1answer
30 views

How to sketch the level curves of $f(x,y) = x^2 - y^2$

I've been practising functions of several variables for college and I've been working with circles all the time $(x^2 + y^2)$, however, I still can't figure out how to solve non circular shapes, as ...
3
votes
1answer
37 views

How to prove that $f=id_A$?

We have a surjective function $f:A\rightarrow A$ and we know that $f\circ f=f$. How do I prove that $f=id_A$?
0
votes
1answer
38 views

Every continuous function $f:[0,1]\rightarrow \mathbb{R}$ is bounded above [closed]

How do I prove, using the Bolzano Weierstrass Theorem, that every continuous function $f:[0,1]\rightarrow \mathbb{R}$ is bounded above? The Bolzano Weierstrass Theorem states that each bounded ...
2
votes
1answer
80 views

Finding inverse of $f(x) =\frac{\ln(x)}{x}$

How do you find the inverse of the following function $$f(x) = \frac{\ln (x)}{x}$$ What looked like a simple question made my head hurt during exam.
0
votes
0answers
8 views

General or generalized, which one for your function name?

When you write a paper in mathematics and you have proposed a generalization of an existing function $F(x)$ like $F(x,y,z)$, you call it generalized F. But, suppose that, in the same paper you, also, ...
0
votes
1answer
18 views

Variations of a sinc function

Consider the following function \begin{align} G(\omega_t) = \frac{1}{N_t}\text{exp} \bigg( j \pi \Delta_t \omega_t (N_t-1)\bigg) \frac{\sin (\pi N_t \Delta_t \omega_t)}{\sin (\pi\Delta_t ...
1
vote
1answer
33 views

If $\int_{0}^{\pi}t^{n}f(t)dt=0$ for all $n\in\{0\}\cup \mathbb N$ then prove that $f \equiv0$

Let, $f:[0,\pi ]\to \mathbb R$ be a continuous function such that $f(0)=0$. If $\int_{0}^{\pi}t^{n}f(t)dt=0$ for all $n\in\{0\}\cup \mathbb N$ then prove that $f \equiv0$. Since, $f$ is ...
3
votes
5answers
56 views

Example $g\circ f=id_A$ but $f\circ g\neq id_B$

Two functions $f:A\rightarrow B$ and $g:B\rightarrow A$. Can someone give me an example where $g\circ f=id_A$ but $f\circ g\neq id_B$?