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

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49 views

Proving that the set of irrational numbers is uncountable [duplicate]

Work: Assume that the set of irrational numbers is countable. Since $Q$ is infinite, it is therefore denumerable. Therefore, there exists a bijective function $f: N \rightarrow Q$. From here I am ...
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0answers
17 views

Domain of function of form $f(x)=\frac{g(x)}{k(x)}$

I just want to know did we have a rule to find the domain of function in form of $f(x)=\frac{g(x)}{k(x)}$ .I know $k(x)\ne 0$ . but in general do we have any rule to compute domain of function like ...
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0answers
26 views

Groups - Compositions

If the f is written to the right of its argument does that mean the composition of $f g$ is actually $g(f(x))$ instead of being $f(g(x))$ which is the notation I'm used to. I ask this because I read ...
2
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1answer
72 views

Pointwise convergence and uniform

There is a norm $\Vert .\Vert$ in the space $C([a,b],\mathbb{R})$ such that convergence pointwise implies convergence in norm $\Vert .\Vert$ ? I think not because if there would be the natural ...
1
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1answer
515 views

Proving the chain rule by first principles

I'm currently trying to prove: $(f(g))'(a)=f'(g(a))*g'(a)$ I have been given a proof which manipulates: $f(a+h)=f(a)+f'(a)h+O(h)$ where $O(h)$ is the error function. However, I would like to have a ...
2
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1answer
244 views

Working out the median of a beta function

I am trying to work out the median of the beta function of $\mathrm{B}(1/2,1/6)$. I have been told the answer to this is $0.9510$ but i'm unsure to get there? Is there a simple formula in order to get ...
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3answers
65 views

inverse of a function $f(x)$, change $x$ and $y$

Find the inverse of the function $f(x)= \dfrac{2x-1}{x^2-1}.$ We switch the $x$ and $y$ letters and then solve the the equation, but it became kind of complicated while solving.
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1answer
63 views

Proving that $f: N\times N \rightarrow N$ is surjective [duplicate]

I am having trouble proving that the function $$f: N\times N \rightarrow N, \ \ f(i,j)=2^{i-1}(2j-1)$$ is surjective. Work: I know that using the theorem in which $n$ is the product of prime numbers ...
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1answer
89 views

Bijective function with different domain and co-domain element count

To be bijective is to be both injective and surjective. Which in other words, have to have a one-on-one match right? Then how am I supposed to come up with a bijective function if the domain has a ...
1
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1answer
73 views

Need helping proving that something is differentiable but not continuously differentiable

I need some help please proving that a function is differentiable at $(0,0)$ but not continuously differentiable at $(0,0)$. The function is as follows... (from $\mathbb{R}^2$ to $\mathbb{R}$) ...
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1answer
31 views

Effect on roots of function on taking the derivative of the function

Suppose there is a function $$f(x)=(x-1)^{15}(x-2)^{20}(x-3)^{25}(x-4)^{30}$$ As we take the derivatives of the function, what will happen to the number of real roots and the number of distinct real ...
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2answers
356 views

Proving that $f$ is a bijection from $N$x$N$ to $N$.

I am having trouble with the following problem: $f: N\times N\rightarrow N$ and $f(i,j)=2^{i-1}(2j-1)$. Prove that $f$ is a bijection thus $N\times N$ and $N$ are numerically equivalent. Work: I ...
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2answers
38 views

Determining if $Z$ is injective or surjective. Help starting a proof

I have $\mathfrak P(\mathbb{R})$ being the set of all subsets of $\mathbb{R}$, meaning $\mathfrak P(\mathbb{R}) = \{X|X\subseteq \mathbb{R}\}$. I then have $F$ being the set of all functions ...
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1answer
44 views

Property of Projection Operator

Let $p: \mathbb{R}^n \rightarrow X$ be the projection mapping onto $X \subset \mathbb{R}^n$ compact convex. I am wondering if $$ \left( p(x) - p(y) \right)^\top z \leq \left( x -y\right)^\top z $$ ...
2
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5answers
695 views

Explanation of recursive function

Given is a function $f(n)$ with: $f(0) = 0$ $f(1) = 1$ $f(n) = 3f(n-1) + 2f(n-2)$ $\forall n≥2$ I was wondering if there's also a non-recursive way to describe the same function. WolframAlpha tells ...
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1answer
133 views

Number of surjective functions from $\{1,2,…,n\}$ to $\{a,b,c\}$

Ok so following questions are given in my text book Let $A = \{1, 2, 3,...., n\}$ and $B =\{a, b, c\}$ then the number of functions form $A$ to $B$ that are onto is. I have no idea how to find ...
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1answer
58 views

How to solve for $f(2)$ given $3f(x)+f(2-x) = 2x^2$?

I just came across the problem where I'm given that $3f(x)+f(2-x) = 2x^2$ and I need to find $f(2)$. I know it is simple, please help me.
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1answer
56 views

Prove that $f$ is a bijection

$f : \mathbb{N} \cup \{0{}\} \to \mathbb{Z}$ $f({}n) = \frac{n{}}{2}$ if $n$ is even $f(n) = -\frac{n{}+1}{2}$ if $n{}$ is odd I want to prove that $f$ is a bijection, and find $f^{-1}$. Now I ...
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1answer
53 views

Isomorphism between rings

Let $R$ be the ring of real valued continuous functions defined on the interval $[0, 1]$. Let $I = \left\lbrace f \in \mathbb{R} : f^2(0) + f^2(1) = 0 \right\rbrace$. 1) Prove that $I$ is an ideal. ...
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1answer
123 views

When does $f_{\omega+1}$ catch up the $G_n$-sequence?

Which is the minimal number k, so that $f_{\omega+1}(n) > G_n$ is true for all $n\ge k$ ? For the definition of $f_{\omega+1}$ look at wikipedia fast growing hierarchy $G_n$ is defined by ...
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0answers
55 views

How many omegas are there in $\large f_{\epsilon_0}$?

For a description look at fast growing hierarchy at wikipedia. $\large f_{\epsilon_0}$ is not defined any more, it is a power tower of omegas, but how many omegas ? I found a defition $$\large ...
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1answer
20 views

Increasing rate of a continuous function

Consider $f: X \rightarrow X$ continuous, with $X \subset \mathbb{R}^n$ compact convex. I am wondering on conditions on $f$ so that there exists $\epsilon > 0$ such that $$ (x-y)^\top \left( f(x) ...
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1answer
24 views

showing that if a function is a bijection, then there exists a an identity function

Let f:x-y be a bijection, show that foi =iof =f where i is identity function. I know that a bijection is one which is bith noe to one and onto. The problems is that the question is so trivial that I ...
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1answer
32 views

Figure out simple joint formula for sets of vectors?

I have on the left side four pairs of sample values that result in the respective pair of values on the right side. ...
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2answers
259 views

How do you shift a sigmoidal curve to the right?

How do you shift the function $1$ $/ ( 1 + e ^ {-x} )$ to the right without altering the shape of the curve?
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1answer
102 views

The difference between semicontinuity and hemicontinuity.

For a point-to-set function F, is "upper hemicontinuous" the same as "upper semicontinuous"? If not, then what's the difference?
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1answer
31 views

Is there a way to compute if(i < j) k := (a + b)c with polynomials over $\Bbb{Z}_p$?

Let $p$ be a prime and let all variables be in $\Bbb{Z}_p$. Then you can write the result of if(i > 0) k = (a + b)c; (C code) as a polynomial $k := ...
2
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0answers
25 views

Quick way to determine the number of horizontal asymptotes

I understand how to calculate horizontal and vertical asymptotes, both by using the trick of comparing the degrees of the numerator/denominator and by using calculus. What I would like to know is ...
0
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2answers
32 views

Inverse Function: unique?

Is it true in general that the inverse of a function is unique if it exists? Why is this so? Clearly inverses in groups are unique. However, that seems not directly applicable in this case...
2
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1answer
96 views

Prove where $|x|^2(\sin(\pi|x|))^2$ (piecewise) is differentiable in $\mathbb{R}^2$

List all points in $\mathbb{R}^2$ at which $f$ is differentiable as well as ALL points in $\mathbb{R}^2$ where $f$ is not differentiable (implied by the first list) when \begin{equation} f(x) = ...
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1answer
28 views

Integration of characteristic function with varying boundaries

I'm a bit puzzled about integrals with indicator/characteristic functions in them. How do I start computing the following integrals? $$ A\int_{-\infty}^{\infty}f(x)\chi_{[-a+x,a+x]}dx $$ and $$ ...
2
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2answers
36 views

Prove that this function is injective

I need to prove that this function is injective: $$f: \mathbb{N} \times \mathbb{N} \to \mathbb{N}$$ $$f: (x, y) \to (2y-1)(2^{x-1})$$ Sadly, I'm stumbling over the algebra. Here is what I have so ...
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2answers
85 views

Rules for combination of odd vs even functional equations

Let $f$ be an even function, and $g$ odd. Let $h$ be some arbitrary function. Is it the case that $f(x) + h(x),\ fh(x),\ hf(x),\text{ and }f(x)h(x)$ are each even or odd according to $h$, and that ...
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2answers
50 views

Finding the Asymptote / Root of a reciprocal function

$$y = \frac{3}{8x - 3} $$ The y-intercept is $-1$ and the vertical asymptote is $x = \frac{3}{8}$ but what would be the horizontal asymptote and the x-intercept in this case? I am asking this as the ...
2
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0answers
124 views

Why is the inverse of the Devil's Staircase not measurable?

I recently did an exercise to show that a monotone function $f:X→ℝ $ is Borel measurable (it even only asked for Lebesgue measurability). On the other hand, the inverse of the Devil's Staircase ...
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1answer
24 views

Functions (Finding Inverse)

$f(x) = x^2 + 2x$ , domain ${x ≥ 1}$ Question: find the inverse The inverse is $f(x) = 1 + \sqrt(1+x)$ (taking the positive square root only) As $f^{-1}(5) = 2$ and as 2 is an element from the ...
0
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1answer
13 views

Functions inverse + domain

Question part a): !([http://imgur.com/sKJbFKu]) Answer: !http://imgur.com/jRfeXkW Can anyone explain why the inverse must be the negative square root?
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1answer
53 views

If $f:[0,1] \to [0,1] $ is continuous, $f(0) =0 $, $f(1) = 1$, and $f^n(x) \triangleq f \ \circ \cdots \circ f(x) \equiv x $, then $f(x) \equiv x$

If a mapping $f:[0,1] \to [0,1] $ is continuous, $f(0) =0 $, $f(1) = 1$, and $f^n(x) \triangleq f \ \circ \cdots \circ f(x) \equiv x $, then $f(x) \equiv x$ The mapping $f$ is injective as $f(x) = ...
7
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2answers
175 views

Prove that no function exists such that…

The exercise goes like this: Find a continous function $f:\mathbb{R} \rightarrow \mathbb{R}$ such that $\forall c \in \mathbb{R}$ the equation $f(x)=c$ has exactly 3 solutions; Prove that no ...
2
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3answers
60 views

Why is the inverse of this function not a function?

Why does $F^{-1}$ need to be defined on all of $Y$? I can have this function: $g(x)=x,\quad x\ne 3$ and even though it is not defined for all $x$ in its domain, it is still a function, right?
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1answer
54 views

Fixed Point Iterations for Bounded Affine Functions

Let $f: X \rightarrow X$ be continuous and $X \subset \mathbb{R}^n$ compact and convex. From Brouwer fixed-point theorem, $f$ admits a fixed point ($\bar{x} \in X$ such that $f(\bar{x}) = \bar{x}$). ...
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1answer
25 views

Something basic; why do I get two different bounds on $f(x) = \frac{x^2}{\sqrt{x^2 + n^{-1}}} + \sqrt{x^2 + n^{-1}}$?

Let $n$ be a natural number. Let $f(x) = \frac{x^2}{\sqrt{x^2 + n^{-1}}} + \sqrt{x^2 + n^{-1}}$. since $x^2 + n^{-1} \geq x^2$, it follows that $$|f(x)| \leq \frac{x^2}{|x|} + \sqrt{x^2 + ...
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2answers
35 views

Problem inverting a function

I have this function: $$v(t)=\sqrt{\frac F c} \tanh \left(\frac{\sqrt{Fc}}{m} t \right)$$ I can visually see that t=6.3 when v=27.8, so why don't I get t=6.3 upon putting v=27.8 in this supposedly ...
2
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1answer
123 views

Intuition - Countable iff Surjection iff Injection [Velleman P310 Thm 7.1.5]

Define $I_n = \{1, 2, ..., n \} $. Let $A$ be a nonempty set. TFAE : (i) $A$ is finite (ie: a bijection $h:A\rightarrow I_{N}$ exists) or A is countably infinite (ie: a bijection ...
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2answers
42 views

Construct a specific function for a given sequence

Given the sequence $A_n=n$, I want to construct a function $f : R \to R$, such that for every $x \in N$: $f(x)=A_x$ $\int\limits_{0}^{x}f(t)dt=\sum\limits_{i=0}^{x}A_i$ How can do that? Thanks
1
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1answer
55 views

Single variable function derivative w.r.t. time?

I was studying calculus and I had doubts about this problem: (this is not homework) A circular wire expands due to heat so that its radius increases with a speed of $0.01 ms^{-1}$. How rapidly does ...
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3answers
34 views

invertibility of $f^{-1}$

In my introductory maths book there is a statement (it follows a theorem) that says:"Note that if f is one-to-one, then $(f^{-1})^{-1} = f$, and so $f^{-1}$ is invertible and also one-to-one because ...
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1answer
37 views

How to establish $\sum_{d|n}d\phi(d)$

I am focusing on #5(b). I do not understand how they go from what I have to the answer. Those are r's at the end.
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1answer
81 views

Before real numbers are precisely defined, $f(x+y) = f(x)+f(y)$ and $f(xy) = f(x)f(y)$… show $f$ preserves order.

Spivak Calculus, 4th ed., problem 3-17: If $f(x)=0$ for all $x$, then $f$ satisfies $f(x+y)=f(x)+f(y)$ for all $x$ and $y$, and also $f(x\cdot y ) =f(x)\cdot f(y)$ for all $x$ and $y$. Now suppose ...
1
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3answers
51 views

Functions definition + question

Am I correct in saying that for Functions, the below is the correct definition: For each value of x in the domain there is only one value of y in the range. Hence, the picture below means that it is ...