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

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2
votes
2answers
125 views

J-measurable sets and functions of class $C^1$

If $f:\mathbb{R}^{n} \rightarrow \mathbb{R}^{n}$ is $C^1$ class and $det f^{\prime}(0)=0$ show that, when $r\rightarrow 0$ $$\dfrac{Vol(f(B[0,r]))}{ Vol(B[0,r])} \rightarrow 0$$ where $Vol(X)$ is ...
1
vote
2answers
34 views

Graphing of Ceiling Functions

How do I graph the function $\lceil x^2\rceil$ (this is ceiling not just brackets). Any explanation is appreciated so I can understand how to!
0
votes
1answer
559 views

Help with Input and Output relationships?

Here's the question: Give three examples of input-output relationships in real life that cannot have negative values in the practical range? Explain why their range cannot have negative values? It's ...
2
votes
1answer
211 views

Proof that a function involving power sets is bijective

I am attempting to solve this following problem, and am having difficulty understanding. The question is as follows: Let $A$ be a set. Let $\phi: \mathcal{P}(A) \to \mathcal{P}(A)$ be defined by ...
1
vote
1answer
31 views

How to do multi-line equations grouped beside one brace

I am not an engineer nor a mathematician. I am stuck as i need to write this equation in a linear method so that i can put it into a VBA program i am working on. I do not understand the order of ...
1
vote
2answers
29 views

Function range problem

$(sin^2\theta +\sin\theta -1)/(sin^2\theta -\sin\theta+ 2)$ . It is asked to find the range of this function. Here i was assumed a variable $z=sin\theta$ so that i get $-1<=z<=1 $ and then ...
-1
votes
1answer
21 views

Looking into mappings

I'm interested in looking at mappings of functions. For example, how would I come up with a function $f:x\in[0,\infty)|\longmapsto\ (-\infty, 0]$ where $f(x)=x^2.$ Basically I want to glue every ...
1
vote
1answer
32 views

real values for a function

We know that $f(x)=ax^2+bx+c=0$ has two real solutions when $b^2-4ac \geq 0$ My question is, if we have a function \begin{equation} f(x)=\frac {D\ln((-0.5\sec x-1)/k)}{\ln a}-\frac ...
0
votes
1answer
98 views

Symbolic rule help

This is a basic algebra question, I could use a little assistant with I just want guidance with symbolic rule. A photocopier was bought for 4000 dollars and depreciates at a rate of $500 per year. ...
1
vote
1answer
140 views

Algebraic methods in determining ONTO and ONE-TO-ONE

I've been looking at ways of determining if a functions is one-to-one or onto. I have a clear sense of this when looking at functions in $\mathbb Z \rightarrow \mathbb Z$ or any other one-dimensional ...
0
votes
1answer
58 views

Proving functions a formal proof

Let $A$ and $B$ be arbitary sets. Let $S_1$ and $S_2$ be arbitrary subsets of A, and let $T_1$ and $T_2$ be arbitrary subsets of $B$. For each of the following state whether it is True or False. If ...
1
vote
2answers
27 views

Function Proof that deals with Set Theory

How to prove that this does not hold if $H$ is not injective or how to show that this equation is true just for injective functions? $$H(X\cap Y)=H(X)\cap H(Y)$$
0
votes
1answer
89 views

Modular arithmetic and one-to-one functions

Let $S = \{0, 1, 2, 3, · · · , 99\}$ . For each of the following functions $f : S \rightarrow S$ , determine whether it is one-to-one and onto, by computing its values for all $k ∈ S$: Function 1: ...
2
votes
1answer
53 views

Composite function intersection

I m stacked in one prove which dealt with sets and functions. I m concerned to prove that: $$f \circ g ( X \cap Y) \subseteq (f \circ g)( X) \cap (f \circ g) (Y)$$ Assume that $g$ is function from $A$ ...
1
vote
5answers
343 views

Existence of a function from $f : \mathbb Z^2 \rightarrow \mathbb Z$

I have a problem with the following question: Does there exist a function $f : \mathbb Z^2 \rightarrow \mathbb Z$ that is one-to-one and onto, and hence invertible? (If yes, then $\mathbb Z^2$ ...
1
vote
2answers
163 views

What is the inverse function of $f(x)=x/(1-x^2)$

Can you give me a hint for how the inverse function of $f\colon (-1,1)\to \mathbb{R}\colon f(x)=\frac{x}{1-x^2}$ looks? I need to show a homeomorphism!
1
vote
1answer
451 views

Finding points of continuity on piecewise function

For what values of $a$ and $b$ is the function continuous at every $x$? $$\displaystyle f(x)=\begin{cases} -1 & \text{if }\;\; x \leq -1\\ ax+b & \text{if }\;\; -1<x<3\\ 13 ...
2
votes
2answers
76 views

Showing Inequality using Gauss Function

If $\alpha, \beta\in \Bbb{R}$ and $m, n\in \Bbb{N}$ show that the inequality $[(m+n)\alpha]+[(m+n)\beta] \ge [m\alpha]+[m\beta]+[n\alpha+n\beta]$ holds iff m=n I thought that we have to ...
-1
votes
2answers
42 views

How to I prove the equivalence of $f(S_1 \cup S_2)$ and $f(S_1) \cup f(S_2)$ (Discrete Mathematics) [duplicate]

If $S_1$ and $S_2$ are both subsets of some arbitrary set $A$, then how do I prove that $f(S_1\cup S_2) = f(S_1) \cup f(S_2)$ for ALL cases I understand that it is true, but I don't know how to prove ...
1
vote
1answer
49 views

How to construct longitudinal from transversal waves and vice versa?

The above construction of a longitudinal wave out of a transversal wave has been encountered somewhere in an old physics textbook. There are several drawbacks with this construction. The maximum ...
1
vote
1answer
64 views

Is $g(x,y) = f(\frac{x}{2},\frac{y}{2})$ correct notation?

I was a bit confused when I saw this statement $g(x,y) = 2f(\frac{x}{2},\frac{y}{2})$, and seeing it used in a double integral $\int \int g(x,y) = 2 \int \int f(\frac{x}{2},\frac{x}{2}) \, dx dy$. I ...
0
votes
2answers
2k views

Difference between minimizing and maximizing functions

Could someone please explain the difference between minimizing and maximizing functions or give me some links to explain the difference in very very very simple terms? I have searched online and I ...
3
votes
0answers
889 views

Left inverse iff injective; right inverse iff surjective

For a function $f:A\to B$, the function $g:B\to A$ is called: a left inverse for $f$ if $g\circ f$ is the identity on $A$ (i.e., $g\circ f = {\rm id}_A$); and a right inverse for $f$ if ...
0
votes
1answer
53 views

Confusion in concept of same function

In this question Let $f(x) = \sin^{-1}(\sin(\tan x))$ and $g(x) = \cos^{-1}(\sin(\sqrt{1-\tan^2 x}))$ are same function, then x belongs to __. What are 'same functions'? No idea how to proceed. ...
1
vote
5answers
124 views

Prove that when $f(A) \subseteq f(B)$ doesn't always mean that $A \subseteq B$

How to prove, when $f(A) \subseteq f(B)$ doesn't "always" mean that $$A \subseteq B$$ when $ f\colon X \to Y $ is total function (not partial)
0
votes
1answer
28 views

Uniqueness of Function (inner product?)

Consider a bilinear function $f : \mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}$ that is symmetric, i.e. $$ f(A_1, A_2) = f(A_2, A_1) \quad\forall A_1, A_2 \in \mathbb{R}^n $$ and satisfies $$ ...
1
vote
2answers
81 views

$Y^\varnothing$ has one element; if $X \ne \varnothing$, then $\varnothing^X = \varnothing$ [duplicate]

From Section 8 of Halmos' Naive Set Theory. (i) Show that $Y^\emptyset$ has exactly one element, namely $\emptyset$. (ii) Show that if $X\neq\emptyset$, then $\emptyset^X=\emptyset$. I'm not ...
0
votes
1answer
22 views

Evaluate a Variable Defined in Terms of its Function

I have a variable x which is defined as follows: x = 150 / (7 + f(x)) where f(x) = 0.005 * x if x > 200, or 100 otherwise. This is actually a simplified version of a real world problem. How do I ...
0
votes
1answer
121 views

What is the rank of the differentiation operator on Pn? What is the kernel?

For this question I was thinking of saying Pn(x)=Ax^n+Ax^(n-1)+...+Ax^2+Ax^1+1 and finding the first derivative P'n(x)=n.Ax^(n-1)+(n-1)Ax^(n-2)+...+2Ax^1+A+0 so in matrix form would get a n by 1 ...
1
vote
4answers
67 views

Defining real powor of linear operator

Let $T$ be invertible linear operator in finite-dimensional vector spaces $V$. How to define $T^a$ for real $a$ such that $T^a T^b = T^{a+b}$ for every $a, b \in \mathbb{R}$?
1
vote
1answer
72 views

What values make this true $f(f(x)) = f(f(f(x)))$ but this not true $f(x) = f(f(x))$

Let $(V,K)$ be a finite vector space, If $f$ is a member of a $\mathrm L(V,V)$ where $V=\mathbb R^2$ and $K=\mathbb R$, what values for $x$ make $f(f(x)) = f(f(f(x)))$ true and $f(x) = f(f(x))$ not ...
1
vote
3answers
90 views

Does $g(f(x))$ imply $g(x)$?

If $g$ is a function of $f(x)$ does this imply that $g$ is a function of $x$? If yes, am I allowed to write the chain rule as: $$\frac{{d[g(f[x])]}}{{dx}} = \frac{{d[g(x)]}}{{d[f(x)]}} \cdot ...
-2
votes
1answer
71 views

Solve $x^4+24x^3+18x^2−27=0$.

Solve $x^4+24x^3+18x^2−27=0$. The answer is quite good looking (for a quartic equation), I don't know much idea. Maybe somebody may give my a bit clue? Thanks.
2
votes
1answer
27 views

An inequality in $L^p$-spaces

Let $\{f_k\}_{k=1}^{\infty}$ be a sequence in $L^p(\Omega,\Sigma,\mu)$ for $1\leq p<\infty$. Suppose $0<c=\inf_k \lVert f_k\rVert_p\leq \sup_k \lVert f_k\rVert_p=C<\infty$ and $f_if_j=0$ for ...
0
votes
1answer
34 views

What does left composition mean in this question?

Consider the vector space of all linear transformations $L(V,V)$ on the vector space $(V,K)$ and a linear map $F:L(V,V)\to L(V,V)$ such that $F(a)= b \circ a$ for all $a\in L(V,V)$, where $b\in ...
3
votes
4answers
106 views

Is the “first nonzero digit” function surjective?

For sets $A= \{x \in \mathbb{R}: 0< x< 1 \}$ and $B=\mathbb{Z_+}$ let $f$ be a function $\space f:A \rightarrow B$ such that $f(x)$ is the position of the first nonzero digit of $x$, ex.g. ...
0
votes
1answer
282 views

If $(V,k)$ is a finite-dimensional vector space, then the space of all linear transformations on $V$ is finite dim and find its dim?

My issue with this is the only way I know how to prove it is to set $\dim V=n$, but then that wouldn't make sense because the second part is find the $\dim$. What I was thinking is using the ...
1
vote
1answer
29 views

Is this possible $(ap+bq)'(t) = (ap)'(t) + (bq)'(t)$, where the prime $'$ denotes differential?

Can $(ap+bq)'(t) = (ap)'(t) + (bq)'(t)$, where the prime denotes differential and $p$ and $q$ are polynomials and $a$ and $b$ are constants. I am asking if the differential of $ap+bq$ with respect to ...
3
votes
1answer
144 views

is this expansion possible (f+g)(t)= f(t)+g(t)?

Hi if the functions were polynomials is (f+g)(t)= f(t)+g(t) possible? I am trying to integrate a function of that form
2
votes
2answers
375 views

Find the matrix A of the linear transformation T(M)

I know that if I substitute the first matrix for $T(M)$ I see what T does to each of the basis vectors. I don't understand how that creates a $3\times 3$ matrix though. I was looking at this ...
0
votes
2answers
50 views

Proof this function is constant

I have the following topological space: $\tau= \{U\subseteq R: 1\notin U\} \cup \{R\}$ and the following application: $f: (R, \tau)\to (R, \tau)$ I have already proved that if $f(1)=1$, then $f$ ...
2
votes
1answer
83 views

show that any continuous function can be approximated uniformly

I do not know where to start because i have not dealt with a question like this before. I feel that i have to use the Stone-Weierstrass theorem, but im not sure how to use it.
2
votes
0answers
181 views

Bounded Function limit proof

Let $ f:\mathbb{R}\rightarrow \mathbb{R} $ be a bounded function and suppose $$g(x) =\sup_{t>x}\hspace{.2cm} f(t)\ .$$ Show that $\lim_{x \rightarrow a^{+}} g(x)=g(a) $ for all real $a$. My ...
1
vote
2answers
45 views

Not injective given cardinalities of sets

How do I prove that a function $f:G \rightarrow H$ is not one-to-one if $|G|=20$ and $|H|=24$?
0
votes
1answer
37 views

Show that $f(E-F) \not\subseteq f(E)-f(F)$ [duplicate]

Can someone help explain this question to me? Let $f : X \to Y$ be a function and $E \subseteq X$ and $F \subseteq X$. Show that in general: 1) $f(E-F) \not\subseteq f(E)-f(F)$ 2) $f(E ...
0
votes
3answers
580 views

center of symmetry formula

How to prove that $I(0,-1)$ is the center of symmetry of the function $$F(x)= x - \dfrac{2e^x}{(e^x -1)}$$ Is there any formula that I can directly apply?
0
votes
1answer
194 views

Prove that the function $f:\mathbb{Q}(\sqrt{2}) \to \mathbb{Q}(\sqrt{2})$ given by $f(a + b \sqrt{2})=a-b\sqrt{2} $ is an isomorphism.

Prove that the function $f:\mathbb{Q}(\sqrt{2}) \to \mathbb{Q}(\sqrt{2})$ given by $f(a + b \sqrt{2})=a-b\sqrt{2} $ is an isomorphism. I need help with this one, mainly proving that f is surjective.
5
votes
4answers
645 views

Showing a function is bijective and finding its inverse

The function f: ℝ2-> ℝ2 is defined by f(x,y)=(2x+3y,x+2y). Show that f is bijective and find its inverse. I've got so far: Bijective = 1-1 and onto. 1-1 if (2x1+3y1,x1+2y1)=(2x2+3y2,x2+2y2) Then ...
1
vote
2answers
55 views

Funtions between sets

If $A$ is a set with $m$ elements and $B$ a set with $n$ elements, how many functions are there from $A$ to $B$. If $m=n$ how many of them are bijections? I got $n^m$ for my first answer. I wasn't ...
0
votes
1answer
38 views

Image definition by Mendelson seems weak

In Bert Mendelson's "Introduction to Topology" image is defined as follows: Let $f: A \to B$ For each subset $X \subset A$, the subset of B whose elements are the points $f(x)$ such that $x \in X$ ...