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

learn more… | top users | synonyms (1)

0
votes
3answers
134 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
69 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
197 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
votes
2answers
266 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
vote
1answer
90 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 ...
1
vote
1answer
48 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 ...
4
votes
4answers
196 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 ...
1
vote
1answer
389 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
votes
2answers
49 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 ...
1
vote
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
votes
2answers
79 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 ...
1
vote
2answers
89 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 ...
4
votes
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 ...
0
votes
1answer
71 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
votes
1answer
273 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
votes
2answers
84 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
votes
1answer
165 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
votes
1answer
84 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.
1
vote
3answers
125 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
21 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
83 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
vote
1answer
44 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
votes
0answers
43 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? ...
1
vote
2answers
129 views

Show that a function $f(x)$ maps to a set of points.Fixed point theorem

Show that the function $f(x)=\frac{1+x^2}{2}$ maps the set of points $0\leqslant x\leqslant 1$ into itself and has a fixed point in that interval even though there does not exists a positive ...
0
votes
1answer
147 views

subadditive function example

Can somebody give me an example of a continuous function $F:\mathbb{R}^2\rightarrow \mathbb{R}$. Such that $$\frac{\partial F(x_1,x_2)}{\partial x_i} \geq 0 \;\; \forall x_i, \;i=1,2$$ but, $$ ...
0
votes
0answers
484 views

Which of the following statements are true about f(x) which is concave down?

For a function $f(x)$ that is concave down (such as the logarithmic or square root functions), and an x-value of $c$, which if the following would be true statements? (There may be more ...
3
votes
1answer
145 views

An n dimensional even function

I am now investigate some behavior of n-dimensional even functions on $\Bbb R^n$. For 1-dimensional even functions, because $f(x) = f(-x)$ for all $x \in \Bbb R$, so we only have to investigate for ...
0
votes
3answers
405 views

What do you call a curve that resembles a crooked S

but it is tilted about theta=artan(x) wit the upper half of the '8' removed about the midpoint and the lower part removed about the of other half. It looks like a tilted 'S' but flipped, and it's ...
1
vote
2answers
149 views

If $f$ is continuous on $\mathbb R$, is $f$ also continuous on $\mathbb R-\{0\}$?

I mean, if I make the latter claim, am I precluding the possibility that $f$ could actually be continuous on the entire reals? (I am right now proving the continuity of a 2-piece function joined in a ...
1
vote
2answers
923 views

Composing two discontinuous functions into a continuous one

Please help me think of an example of two discontinuous functions on $\mathbb R$ whose composition gives a continuous function on $\mathbb R$.
-2
votes
2answers
536 views

Change parabolic equation to canonical form

I have equation $y = -x^2 + 2x + 7$. How can I change it to canonical form, which looks like $y^2 = 2px$ ? ($p$ will be parameter) What i ve tried so far: $$\begin{align} y &= -x^2 + 2x + 7\\ y ...
3
votes
1answer
226 views

Prove the Borel Lemma

I'm trying to prove the Borel Lemma, which is: For every series $a_0,a_1,a_2,\dots$ in $\mathbb{C}$ exists $f \in C^{\infty}(\mathbb{R})$ such as $$ f^{(k)}(0) = a_k $$ for every $k \in ...
4
votes
3answers
110 views

Bounds for solutions of $xe^{x} -n =0$ for $n\geq 3$

$\alpha _{ n }$ is a solution for $f_{ n }=xe^{ x }-n =0$. How can I prove that $\forall n\geq 3$, $\ln(n)-\ln(\ln(n))\le \alpha _{ n }\le \ln(n)$
1
vote
3answers
124 views

confusion regarding the concept of a function

I was reading principles of mathematical analysis by Walter Rudin chapter 2 when a confusion about the definition of a function cropped up. (Read definition in comment below) I had thought that ...
2
votes
1answer
108 views

Continuity and Differentiability

Prove that if $f:\mathbb R\to\mathbb R$ is continuous at $a$ and differentiable at all $x\neq a$ in a neighborhood of $a$, and $\lim_{x\to a}f'(x)=L$, then $f$ is differentiable at $a$ and $f'(a)=L$. ...
2
votes
3answers
161 views

Inverse of inverse of function?

What is the inverse of inverse of a function (I assume the original function is invertible)? Is this the original function? Is it always true?
7
votes
1answer
282 views

Intuition behind convex functions

For me, possibly the most out-of-nowhere definition of the first semester of Calculus was the following definition of a convex function and its equivalents. Function $f$ is convex on the interval ...
0
votes
1answer
38 views

An upper bound?

Let $\alpha = n^{c_1}$ and $\beta = n^{c_2}$. Is there a good upper bound (an asymptotic bound would be good as well) for $f(n) = (1 - \frac{1}{\alpha})^\beta$. I am particularly interested in a bound ...
0
votes
1answer
41 views

When is $f(x) = \frac{ax+a}{e^{-b}-{(e^{c})}^x}$ monotonic? [duplicate]

Possible Duplicate: When is $f(x) = \frac{ax}{e^{-b}-{(e^{c})}^x}$ monotonic? When is $f_2(x) = \dfrac{ax+a}{e^{-b}-{(e^{c})}^x}$, where $(a,b,c)$ are positive real numbers, monotonic? ...
1
vote
3answers
111 views

When is $f(x) = \frac{ax}{e^{-b}-{(e^{c})}^x}$ monotonic?

Please consider the function $f(x) = \frac{ax}{e^{-b}-{(e^{c})}^x}$ where $(a,b,c)>0$ are positive reals. For what values of $x$ is $f(x)$ monotonic? For example, is this true for $x>0$? ...
1
vote
1answer
64 views

Function for unique hash code

I am interested in finding $F(x,y)$, such that $x$ and $y$ $\in \mathbb Z^+$ and $F(x,y)$ is one to one function i.e., $F(x,y)$ is unique for any unique unordered pairs of $x$ and $y$. Regards, ...
2
votes
1answer
228 views

If $f^{-1}$ has nowhere zero derivative, then $f$ is differentiable

I think you need to apply theorem 5 from chapter 12 of Spivaks calculus, but not sure how to. If someone could please help me. The question is: Suppose $f$ is a one-one function and the inverse ...
0
votes
1answer
34 views

If $\hat{f}(a)=p^{-1}\sum_t f(t)\psi_{-a}(t)$, why does $ f(t)=\sum_a \hat{f}(a)\psi_a(t)$?

Suppose $f\colon\mathbb{Z}\to\mathbb{C}$, and let $p$ be a prime such that $f(n+p)=f(n)$ for all $n$. I denote by $\psi_a(t)=\zeta^{at}$ where $\zeta$ is a primitive $p$th root of unity. It might be ...
1
vote
2answers
193 views

Probability distribution function of the length of an interval taken from a uniform probabilty distribution.

This is a consequence of my suggested solution to this question. Consider the probability distribution function that is uniform over the interval $[-a,a]$: $$F(x)=\begin{cases} 0 & x \leq -a\\ ...
5
votes
3answers
136 views

Calculate $\lim_{x\to 0}\frac{1}{x^{\sin(x)}}$

Calculate $$\lim_{x\to 0}\dfrac{1}{x^{\sin(x)}}$$ I'm pretty much clueless here, only that there is L'hospital obviously here. Would appreciate any help.
3
votes
1answer
260 views

Proof with function composition

Prove or disprove the following statement: Function $f: S \rightarrow S$, where $S$ is non-empty, is bijective if and only if there exist unique functions $g, h : S \rightarrow S$ such that $$ ...
6
votes
2answers
2k views

Bijection from (0,1) to [0,1)

I'm trying to solve the following question: Let $f:(0,1)\to [0,1)$ and $g:[0,1)\to (0,1)$ be maps defined as $f(x)=x$ and $g(x)=\frac{x+1}{2}$. Use these maps to build a bijection ...
3
votes
3answers
8k views

The relation between an exponential function and a logarithmic function

I have been told multiple times that the logarithmic function is the inverse of the exponential function and vice versa. My question is; what are the implications of this? How can we see that they're ...
4
votes
2answers
225 views

Existence of an infinitely differentiable function $ f $ with $ {f^{(n)}}(0) = 0 $ for all $ n \in \mathbb{N} $.

How can one show that there exists an infinitely differentiable function $ f: \mathbb{R} \to \mathbb{R} $ such that $ {f^{(n)}}(0) = 0 $ but $ f^{(n)} \not\equiv 0 $ for all $ n \in \mathbb{N} $?