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

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205 views

Approximate a complicated mystery function

Let there exist a mystery function ƒ. ƒ accepts exactly 2 arguments, A & B. As B approaches A, ƒ approaches A, at a simple exponential growth rate E. As B approaches 0, ƒ approaches the mean ...
1
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1answer
40 views

If there is a $T$ such that $V(t)<V(t-T) \ \forall t$, does that imply $V(t) \to 0$?

Let $V(t)$ denote a continuous scalar function $\mathbb{R} \mapsto \mathbb{R}$. Assume that we can find a constant $T \in \mathbb{R}$ such that $V(t)<V(t-T)$ for all $t$. Does that imply that $V(t) ...
3
votes
2answers
109 views

Average of function, function of average

I'm trying to find all functions $f : \mathbb{R} \to \mathbb{R}$ such that, for all $n > 1$ and all $x_1, x_2, \cdots, x_n \in \mathbb{R}$: $$\frac{1}{n} \sum_{t = 1}^n f(x_t) = f \left ( ...
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1answer
84 views

Looking for a counterexample [duplicate]

Possible Duplicate: Solution(s) to $f(x + y) = f(x) + f(y)$ (and miscellaneous questions…) I am looking for a function $f:\mathbb{R}\rightarrow \mathbb{R}$ that for all $x$ and $y$ ...
2
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1answer
65 views

Recurrence relation help

The function $\psi_k(n)$ satisfies the recurrence relation: $$\sum_{j=0}^k\binom{k}{j}(-1)^j\psi_j(n)\ln(n)^{k-j}=\psi_k(n)$$ Using this, is there a general way I can re-write the function $ ...
0
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2answers
56 views

For what $f(n)$ does $O(f(n) \log n)=O(\log\log n)$?

$k=f(n)$. Given $O(k \log_2 n)$, what function $f$ of $n$ would be needed for it to equal $O(\log_2 \log_2 n)$? (where $k \in n \in \mathbb{Z}^+$)
3
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2answers
223 views

Question about bijective function when restricting domain

Let $f(x):x^3-x$. By an appropriate restriction of the domain and range to find a bijective function $g$. Then graph $g$ and $g^{-1}$. The function I found is ...
3
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1answer
450 views

Drawing by lifting pencil from paper can still beget continuous function.

From page 105 of the 1994 edition of Spivak's Calculus: A continuous function is sometimes described, intuitively, as one whose graph can be drawn without lifting your pencil from the paper. ...
4
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4answers
1k views

If $f(x)<g(x)$ prove that $\lim f(x)<\lim g(x)$

I have this question: Let $f(x)→A$ and $g(x)→B$ as $x→x_0$. Prove that if $f(x) < g(x)$ for all $x∈(x_0−η, x_0+η)$ (for some $η > 0$) then $A\leq B$. In this case is it always true that $A ...
0
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1answer
58 views

asymptotic expansion, interpretation

I am interested in asymptotic behavior of a function at infinity: $$ f(r)=\frac{0.04962 e^{-2 r} (r-1.000)}{\left(\left(e^{-2 r}\right)^{2/3}+0.06119\right)^2 r} $$ Tried ...
3
votes
3answers
104 views

Show that the only function $f$ satisfying this property is the constant function $f(x) = a$

Let $f : (−1, 1) \rightarrow R $ be a continuous function with the property that $f(x) = f(x^4)$ for all $x$ and $f(\frac{1}{2}) = a$. Show that the only function $f$ satisfying this property ...
1
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4answers
175 views

Transitivity of 1-to-1 functions

If $g:A\rightarrow B$ and $f:B\rightarrow C$ are both 1 to 1, how do we prove that $f\circ g$ is 1-to-1? What type of a proof method would be required exactly?
4
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0answers
42 views

Symbol for function composition [duplicate]

Possible Duplicate: History of $f \circ g$ Choice of symbols can be an indicator of intellectual allegiance. Consider how, back in the day (and before LaTeX regularised things so much!), ...
4
votes
1answer
221 views

Function that is discontinuous only for integer fractions

I have this question: Find a function $f :\mathbb R \to\mathbb R$ which is discontinuous at the points of the set $\{\frac1n : n \text{ a positive integer}\} \cup \{0\}$ but is continuous ...
0
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1answer
66 views

Basic limit terminology question

If the left and right limits of a function $f$ differ, do we then say that "$f$ has no limit at $c$"? (This sounds wrong considering that f has both left and right limits... but we obviously can't ...
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2answers
50 views

What is the smallest value of $n$ for which the maximum of $f_n(x)$ occurs when $x = 3$?

The function $f_n(x)$ is defined by the equation $f_n(x) = x^{n–x}$, where $0 \le x \le n$ and $x$,$n$ are integers. What is the smallest value of $n$ for which the maximum of $f_n(x)$ occurs when $x ...
3
votes
1answer
1k views

Lim Sup/Inf for real valued functions

To understand the notion of, say, limit superior for a sequence, is not difficult. Simply consider the set of all upper buonds for the set of all limit points of the sequence, and then simply pick the ...
0
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2answers
179 views

How to prove this surjectivity?

Let $f:\mathbb{R} \to \mathbb{R}\times\mathbb{R}_{0}^{+} \space$ be defined as $f(x)=(x+1,x^2)$. To prove that this function is surjective I started by the definition. $$\forall \space (a,b) \space ...
2
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3answers
167 views

Is the function $f:\mathbb{R}^2\to\mathbb{R}^2$, where $f(x,y)=(x+y,x)$, one-to-one, onto, both?

I have to determine whether the function $f:\mathbb{R}^2\to\mathbb{R}^2$ defined by $f(x,y)=(x+y,x)$ is one-to-one, onto, and if both then describe its inverse. I'm relatively new to mapping since ...
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1answer
192 views

Statements about a twice differentiable function

Can you help me to prove or disprove? We have a function $f:(0,+\infty)\rightarrow\mathbb{R}$ twice differentiable, such that as $x\rightarrow+\infty$ (a) $xf(x) \rightarrow+\infty$ (b) $xf''(x) ...
1
vote
2answers
189 views

How do I find the delta analytically for $f(x)$ with a degree other than $1$

So I know how to find the $\delta$ in $f(x)$ that can be factored into a degree of $1$, or I can solve it by solving $L + \epsilon = f(x) $ then finding the distance from $a$. But how do I find the ...
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1answer
90 views

Showing two functions are bijections

I would like to verify my proofs for the following problems. (a) Show that for $n > 1$ there is a bijective correspondence of $A_1 \times A_2 \times ... \times A_n$ with $(A_1 \times A_2 ...
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1answer
86 views

Norm of normal Operator A

I just found the following equality $||A||=sup_{\lambda\in\sigma(A)} |\lambda|$ My question: What exactly means $\sigma(A)$ and why this is true ? I always thouht the only way to get the ...
6
votes
3answers
182 views

Show that $\frac{f(n)}{n!}=\sum_{k=0}^n \frac{(-1)^k}{k!}$

Consider a function $f$ on non-negative integer such that $f(0)=1,f(1)=0$ and $f(n)+f(n-1)=nf(n-1)+(n-1)f(n-2)$ for $n \geq 2$. Show that $$\frac{f(n)}{n!}=\sum_{k=0}^n \frac{(-1)^k}{k!}$$ ...
0
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1answer
740 views

How to prove this limit composition theorem?

If $$\displaystyle \lim_{x \rightarrow c}f(x)=l$$ and $$\displaystyle \lim_{x \rightarrow l}g(x)=L$$ and $f(x) \neq l$ in some punctured neighbourhood of c, then $\displaystyle \lim_{x ...
0
votes
2answers
325 views

Functions-Set Theory Proof that $f(C \cup D) = f(C) \cup f(D)$ [duplicate]

Possible Duplicate: Prove $f(S \cup T) = f(S) \cup f(T)$ I'm revisiting set theory and am troubled by this question. Let $f:A \rightarrow B$, and $C \subset A$, $D \subset A$. Prove that ...
0
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3answers
115 views

Intermediate Value Theorem guarantee

I'm doing a review packet for Calculus and I'm not really sure what it is asking for the answer? The question is: Let f be a continuous function on the closed interval [-3, 6]. If f(-3)=-2 and ...
5
votes
1answer
68 views

How to show $f(x) \leq 1+\frac{\pi}{4}$ for every $x \geq 1$

Suppose $f$ is a real-valued differentiable function defined on $[ 1,\infty)$ with $f(1)=1$. Suppose , moreover , that $f$ satisfies $$f'(x)=\frac{1}{x^2+f^2(x)}$$ Show that $f(x) \leq ...
3
votes
1answer
189 views

$f$ is continuous at $c$ $\implies$ $f$ has a limit at $c$. True?

Further to Another simple/conceptual limit question where I was questioning David Brannan's assertion in his A First Course in Mathematical Analysis that $f(x)=\sqrt x,x\geq 0$ has no limit at $0$ ...
2
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2answers
233 views

Find the equation of this bijection from $ \mathbb{R} $ to $ (0,1) $.

I need some help (hints or an answer) in finding the actual equation of this bijections from the reals $ \mathbb{R} $ to $ (0,1) $. We may assume that the radius of the circle is $ \dfrac{1}{2} $. ...
1
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1answer
86 views

Special case of combinatorial onto functions

Let $[n]$ be the set of integers $\{1, 2, \ldots, n\}$. I want to find the number of onto functions from $[m]$ to $[3]$. The answer I found was $3!\cdot (3)^{m-3}$ My reasoning is: We have a ...
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1answer
47 views

Combinatorial Correctness of one-to-one functions

Let $\lbrack k\rbrack$ be the set of integers $\{1, 2, \ldots, k\}$. What is the number of one-to-one functions from $m$ to $n$ if $m \leq n$? My answer is: $\dfrac{n!}{(n-m)!}$ My reasoning is the ...
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4answers
195 views

Question on functions and derivatives

I can't seem to get this subject very well. Let $f(x)$ be twice differentiable on $[0,1]$, and that there is a constant $A$ so that $|f''(x)|\le A$. Show that if $f(0)=f(1)=0$, then $|f'(x)|\le ...
0
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1answer
360 views

Show that there exist positive constants $a$, $b$ such that $|f(x)| \leq a |x| + b$ for every $x \in \mathbb{R}$.

Problem For a uniformly continuous function $f:\mathbb{R} \rightarrow \mathbb{R}$, show that there exist positive constants $a$, $b$ such that $|f(x)| \leq a |x| + b$ for every $x \in \mathbb{R}$. ...
0
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2answers
47 views

A question on the relations between a function and its derivatives.

Got stuck on this one - Let $f:\mathbb R \to \mathbb R$ be differential twice. Show that if there are three different solutions to the equation $f(x)^2=x^2$ then there is at least one solution to ...
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1answer
25 views

Quantification of differences between distributions

I am trying to find a method that will allow me to quantitatively differentiate between 2 distributions. The distributions show a peak where there is positive alignment in a certain direction and ...
2
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2answers
76 views

Another simple/conceptual limit question

Further to my previous post Very simple limits question to clarify my understanding , here's a related question. Let $f(x)=\sqrt x,x\geq 0$. What is the limit of $f$ as $x$ tends to $0$? I think the ...
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2answers
88 views

Derivative for log

I have the following problem: $$ \log \bigg( \frac{x+3}{4-x} \bigg) $$ I need to graph the following function so I will need a starting point, roots, zeros, stationary points, inflection points ...
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3answers
1k views

Why is $y = \sqrt{x-4}$ a function? and $y = \sqrt{4 - x^2}$ should be a circle

So why is it a function, even though for example $x = 8$; you'll have $y = +2$ and $y = -2$. It'll fail the vertical line test. But every textbook considers it as a function. Did I misunderstand ...
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1answer
67 views

Very simple limits question to clarify my understanding

$f$ is a piecewise function defined as $f(x)=x$ if $x\not= 2$, and $f(x)=5$ if $x=2$. What is the limit of $f$ as $x$ approaches $2$? Is the answer $2$ or $5$? I'm guessing the answer is $2$ but am I ...
0
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1answer
249 views

Proper representation for a conditional function

I have a function shown in below image. But the 2nd line (the parenthesis) has something that I don't know how to formulate properly. The equation (I was not allowed to post images directly) 1- Does ...
0
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2answers
82 views

Give an example of a map $\pi:\Bbb N\to\Bbb N$ with a right inverse but no left inverse and vice versa

I'm really struggling with this as I can't think of anything that would work one way and not the other
3
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1answer
163 views

Functions in calculus - notation

I don't have an extensive formal training in calculus, but I'm doing quite a lot of differential calculus work at the moment and there's something which bothers me. Say I have the differential ...
0
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1answer
79 views

What is the domain of $f(x) = \ln(1+x)^{\sin(x)}$?

Wolfram Alpha says that the domain of $$f(x) = \large \ln(1+x)^{\sin(x)}$$ is $x \gt 0$. I can't figure out how we come up with this restrictions. Please help.
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3answers
112 views

Function which gradually rises until some point and then quickly “falls”

Could someone point me to any function ${ f(x) }$ which is continuous at some interval ${ x \in [x_0; x_1] }$ and can be represented by formula, so that it rises until some point and then quickly ...
2
votes
1answer
20 views

Estimation for large $k$.

I have a function $f(k)$ defined on the set of natural numbers and I managed to show that $f(k)>n-\binom n k(1-n^{-2/3})^{k(k-1)/2}$ for all integers $n\ge k$. I am hoping to get a further ...
10
votes
3answers
3k views

Is composition of measurable functions measurable?

We know that if $ f: E \to \mathbb{R} $ is a Lebesgue-measurable function and $ g: \mathbb{R} \to \mathbb{R} $ is a continuous function, then $ g \circ f $ is Lebesgue-measurable. Can one replace the ...
2
votes
2answers
79 views

Simple bijectiveness question

If $f$'s image set is $g$'s domain and vice versa, does that imply that their domains have a 1-1 correspondence? That $f$ and $g$ are both bijective mappings? Are my questions even meaningful? Edit: ...
1
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1answer
42 views

Complex number second degree function

This is my first question posted here, but I came across this when following an example in my textbook. It's part of factorizing an equation to enable a Laplace Transformation. $s^2 + 4s + 5 ...
0
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0answers
42 views

What are the features of $n/d, n \rightarrow d, d \rightarrow \infty; n, d \in \mathbb{N} $?

What are the features of $n/d, n \rightarrow d, d \rightarrow \infty; n, d \in \mathbb{N} $? What is the value of $\lim_{n \rightarrow d, d\rightarrow \infty} (n/d)$? What is the function's range? ...